The plant doesn't look unhealthy ... just has a couple of broken leaves. If you like, you can cut them off at the stem, using a clean, sharp knife, but the plant is fine as it is.
I can't tell for certain which variety your aloe is, but it looks like it might be the darker green variety, which is not nearly as prolific at producing pups as the lighter green/variegated/speckled type. You should still get some pups, though. It's very possible that your plant just isn't old enough yet to produce pups.
Aloes, like cacti, usually produce more pups when they are slightly root bound. If you repot your plant, be careful not to put it in too large of a container. Although aloes are adaptive to a wide range of conditions, they do best in a potting mix designed for cacti and succulents.
They can survive in low light conditions, but they prefer a lot of sun. During the winter months, you might want to consider using grow lights to increase the light intensity for your plant. You can move your plant outdoors in the summer, but don't put it into direct sunlight right away. Start it in shade, then every couple of days, slightly increase the amount of sunlight it gets.
"Isn't it enough to see that a garden is beautiful without having to believe that there are fairies at the bottom of it too?" - Douglas Adams