The problem that keeps it from starting is probably just lack of compression due to the stuck rings. I had misread your post when you said the rings are not broken - I thought you had said they are broken. Stuck rings cause fairly massive blow-by, carboning up of everything, scoring of the piston, and pretty soon, scoring of the bore as well. Do not run it with stuck rings.
If the rings are stuck but the piston scoring is not bad, I'd try to remove the rings without breaking them, clean out the ring grooves - carefully - as well as possible with a hand-made scraper, and try the old rings in the bore to check the ring gap. The gap will probably be bad: that will be what caused the blow-by that caused the carbon build-up that caused the rings to stick. If the gap is big, you need new rings. Unless the piston scoring is bad, that is all you need. Clean the ring grooves, fit new rings, put it back together. If you want to be thorough, check the gap in the new rings by trying them in the bore - but it should be OK, even if not perfect.
Old guy's trick: once you have verified that you can't re-use the old rings because the gap is really outrageous, break one of them, grind the broken end square, and use it for a scraper to clean out the ring grooves.
