Hire a contractor to get rid of the tree roots.
Rent a tiller and go over the whole garden area. That should deal with the ivy, and aerate the soil.
Dump in a zillion quarts of peat moss. Spade it in.
Ditto manure.
Finally, without knowing where you are and what kind of soil you have, making suggestions on lawns is pretty tough. The best bet is to contact your local State (or County) Agricultural Extension Service for specific advice on the best local grass seed to use.
Sorry, I don't know anything about Devil's Ivy, but have fun! It's a lot of work, but the payoff is well worth it. Do a little bit at a time so you don't burn out.
Chris