If the wheel is so far out of true that the rim cannot be bent back into shape with spoke tension alone then the only option is cold setting (bashing rim against work bench) to straighten things out, which can weaken the wheel quite a bit as aluminum (most likely your rim material) has no elasticity. Also, replacing spokes should involve detensioning, putting in new spokes, and then tensioning the wheel. This would probably take 15-20min for a good mechanic and up to an hour for a less experienced mechanic. This would incur a cost in the ~$30 range and a new wheel freewheel wheel with aluminum rim was about $50 when I used to work in shops so the new wheel might be your best option considering the finite lifetime of aluminum rims and the fact that you are breaking spokes.
If you could do the work yourself I would say to go for it, but if you have to pay labor costs that are much greater than parts that I get the feeling that you will soon be throwing good money after bad.