I am a stay at home mom with a 4 year old and 1 1/2 year old. I struggle to keep the house clean, the children tear it apart in two seconds flat. That being said, I do make sure to have dinner ready most nights by the time my husband is home from work. I do all the laundry, bill paying, grocery shopping, and most of the cleaning.
It's nice your husband is able to stay home with the children. It isn't easy! But he does need to pull more weight in the housekeeping department. I would also recommend counseling (especially if your work has an employee assistance program), and discuss expectations of how you both contribute to the family. You definitely shouldn't be working all day and doing the housework and cooking!
Is your husband depressed? There are some cool free courses on coursera.org your husband could check out and see what he's interested in. Perhaps he can do some online courses so he can be prepared to get a career in a few years? It might be nice for him to have something to work towards.