What would it take to balance a stereo power amp ?

There's really nothing you can do to balance the amp if it was designed single ended. Your only choices are to continue to run it unbalance, or replace it.

There is a way & it's very simple to accomplish!
All you need to do is install some balancing transformers on the inputs, assuming you can solder & you have basic electronics skills? Steve McCormack has these parts available. Very likely you can also get them from Michael Percy Audio. A local musician instrument shop is another likely source if you're not seeking anything too esoteric (I wouldn't - based upon the context of your post).

I agree completely with Bob Bundus. It's easy, cheap, and done all the time in pro-gear setups. They even have adapter plugs which have the balancing transformers built right into them, and have the XLR on one end and the RCA on the other.

Since the balancing question has been addressed, let me mention some other possibilities that might cause hum & noise.

Having had experience running sound I'd like to throw a few suggestions out. You may already know this but here goes anyway.

Make sure all grounds are grounded, including mics. Using 57's & 58's? Conversely, make sure things that aren't supposed to be grounded aren't, like those 1/4" jacks. For instance, the little plastic insulator has a way of disappearing.

Same goes for all the cables & the snake. Additionally, only use shielded for shielded applications, etc.

Make sure power is in phase. This will help the most when you plug in, as over half the places you'll set-up in will have screwed up AC. If your stuff is oriented correctly, easier to adjust. You probably don't have to now but learn how to get power from panels. Bad AC, dimmer switches & fluorescent lights are the PA's biggest enemy.