I'll bet your rear hub is pretty well worn out. That's probably where your slippage is, and also lack of braking. The first order of business is to assess the rear hub. Take it apart and see if it just needs cleaned up and re-lubed and adjusted. It's old if it's a skip tooth so find the name brand on the brake arm and find help with that particular hub. Single speed hubs aren't that complicated. Here's a New Departure breakdown:
Then, if it's beyond help and worn out, a new wheel and hub won't be skip tooth unless you invest some bucks. If you're just wanting to ride, I would suggest conversion to standard chain. You'll have to change out your chain wheel also, but that's not hard to do either. New 26" back wheel, chain, and chain wheel, and you'll be cruising. Save all the old stuff in a box.