It took me longer then I thought as I got so tired I had to stop. But here we go!
Received this response in a pm and am posting it here for continuity.
Thanks for responding. We are in St. Louis which I believe is zone 6. The back of the house faces north and is a wooded area that slopes down to a creek. Close to the house, there are about 8 tall trees with the ivy. Three years ago, a few very large sick trees were cut down. Since then the shady area changed to a partly sunny area. At that time, the ivy started dying where the trees were. The ivy continues to die at a slow pace, creeping across the area. I definitely think that sun exposure caused by the trees disappearing caused the dying to start. This past spring I cut down 3 overgrown yews. Underneath them, the ivy looked very healthy. After the ewes were cut down, the ivy died. Now, half the area has healthy ivy and half has dead brown vines. Any help on what to do would be greatly appreciated!!!!!
Here's what I would do. Hand digging is always an option but you might want to consider using vinegar. Household vinegar is 5% acidic and horticultural vinegar is 15% to 20% acidic. Vinegar is used as you would any non-selective herbicide as it's sprayed on the leaves of weeds. With woody weeds such as ivy you would need to soak the vines in the vinegar when it's in active growth and has leaves. You may have to wait for spring to do this but if it's still warm where you are, give it a go and see what happens with a vine or two. I would suggest you try some full strength household vinegar for this. I used this method to get rid of trumpet vine - campsis radicans that has a similar root invasion habit to other vines like wisteria and ivy before I knew about vinegar.
When I moved into this house 19 years ago as a renter, I inherited a trumpet vine - campsis radicans. Here's my horror story and what I've learned about this vine. Over time the vine began to bloom and pop up everywhere in the yard. I would pull the sprouts only to find more year after year. When it pops up in the lawn it can just be mowed. After 13 years we purchased the house and had to cut down 5 trees and regrade the land due to overplanting and flooding. When we dug up the stumps from the trees and regraded we discovered roots of the vine 3' to 4' deep in the soil, up to 30' from the parent plant and as large around as my wrist! We dug and dug and, well you get the point. A year later we still had sprouts coming up from bits of roots that we'd missed.
Here is how I've learned you can get rid of it. Now, up until this point I had NEVER used herbicides or pesticides in the garden. Here's what I did and you can do to get rid of it. Put about an inch of Round Up Weed and Grass Killer Super Concentrate (you could also use Brush B Gone) in a clear plastic container with a tight fitting lid like you might get at the deli with potato salad. Cut a slit in the lid and insert the tips of the vine in the solution when in active growth (has leaves on it and the leaves need to be in the solution). Leave the vines in the solution for 48 hours and then cut the vines near the lid. To remove the vine from the lid, be sure and take the container to a safe place so that no solution splashes on anything precious. You can reuse the solution until it is all absorbed. Every time I find a new sprout I do this same procedure. So far there have been no sprouts from areas that were treated this way.
Now what to replace the ivy with. Since this is part sun, on a slope leading to a stream and you have kids, I would suggest some native groundcovers or non-natives that aren't invasive in the environment. These won't take over, will help to hold the soil in place and offer berries and nectar to wildlife. For a large area you could select 3 different ones so it won't look too busy. Using mostly evergreens would probably be best in your zone so you'll have something to look at in winter when there isn't snow cover. Everything here is carefree once established. Here's some ideas. All are evergreen unless noted.
Arctostaphylos uva-ursi 'Point Reyes' aka bearberry aka kinnikinick. Probably best in a sunnier spot and one you can more easily weed if needed.
Carex glauca aka Blue sedge. It's not native but isn't invasive and will give the look of grass in areas you would like to see a grassy look.
Carex pennsylvanica aka Pennsylvania sedge wouldn't make the best groundcover to suppress weeds, but could be very valuable for holding the soil and for wildlife near the stream.
Chrysogonum virginianum aka Goldenstar aka Green and gold can be a bit fussy about where it likes to grow, but once it finds a spot that makes it happy, it's lovely. It doesn't like wet leaves on it in winter and summer drought will send it shriveling up. Once it gets water it springs back. I used to grow this but lost it during a the major renovation above. I did share some with a neighbor and it's still happily growing in a sunny spot under his south facing shrubs. It may not stay evergreen in a cold winter for you, but it did in my zone 7 garden.
Gypsophila repens aka Creeping baby's breath isn't native and will do best in a sunny spot in a rocky area, but would be a lovely accent in a few well placed spots. It isn't evergreen and there are pink flowered varieties available as well.
Houstonia serpyllifolia aka Creeping bluet aka Thymeleaf bluet is a native that will need a moist shady spot and looks alot like moss when not in bloom.
Koeleria cristata aka Junegrass aka Koeler grass aka Prairie junegrass is a native that is great for erosion control once established. It will seed around a bit and you'll find it as a happy surprise. It grows in clumps and is great for wildlife. The flowers are tiny and on spikes. It is not evergreen.
Laurentia fluviatilis aka blue star creeper is not native and probably won't be evergreen in your zone.
Mazus reptans aka Creeping mazus isn't native and probably won't be evergreen in your zone, but it should bloom on and off for a long period of time and is good at suppressing weeds.
Microbiota decussata aka Russian arborvitae aka Siberian cypress is one of my favorite evergreen conifers. When you get sick of seeing junipers and getting scratched by them, this is the one to plant. It will spread very wide (up to 12 feet) so site it well, but it will hold a hillside together and has soft foliage.
Here's how it looks in spring.
Phlox subulata aka Moss phlox aka Moss pink is a native that will need full sun. It's flowers come in different shades of either pink, blue or purple. There are several types of phlox so choose by scientific name.
Rhus aromatica 'Grow-Low' aka Grow Low fragrant rhus aka 'Gro-Low'. This is the dwarf selection of this native plant so you'll have to purchase it by specific name. Gorgeous fall color. It is not evergreen.
Asarum canadense aka Canadian Wild Ginger is a native groundcover for the shade that will gently spread and looks wonderful under trees. It even looks great with ferns growing through it. It is not evergreen. There is a non-native woodland ginger that has shiny leaves, but I like the native one better.
Isopyrum biternatum aka false rue anemone is a lovely native for the shade that isn't evergreen.
Mitchella repens aka partridge berry is a native evergreen woodland groundcover with red berries in the winter.
Gaulheria procumbens aka wintergreen aka teaberry is another wonderful native evergreen woodland groundcover.
If you want to drool over some plants that grow in your region you can spend a sleepless night here. Not all are natives and some are invasive but you can scroll to the bottom of the page for the listing of all the plants and use it for research if you like.
If you think you might like to add ferns to the shadier areas, this site has a nice list of natives.
My favorite would be one of the Maidenhair ferns. Adiantum pedatum aka Northern Maidenhair fern. This native fern would look stunning planted on the hillside or near the stream.
Even better would be Adiantum capillus-veneris aka Southern maidenhair fern aka common maidenhair fern planted near the water. This wouldn't be evergreen or a groundcover for you really, but just something lovely to dress up an area.
I know I've thrown alot at you, but you can research any of these and others you find at google. You can even click on 'Images' to get lots of pictures.
If you decide to mailorder any plants you can check references here. You can even search by state and plant material.
Don't hesitate to ask more questions.