There really isn't any good way, that I know of, to properly bleed the brakes without a working bleeder valve. You've got air in the system now, so either you will have to repair the broken bleeder valve or replace the caliper.
Try an "Easy out" before you go shopping for another caliper. Any auto parts store should have them. Be careful of the cheap ones, the tip will snap off if your not careful. Get a good set if you can. They will pay for themselves after a couple of uses. If you can find one small enough to fit the existing hole. If you can, remove the caliper, disassemble and put in a bench vise. If no bench vise, you may have to do the operation while it’s attached to the bike. Either way, lightly tap in the "easy out", and give it a twist with the appropriate wrench. "PB Blaster" or "Liquid Wrench" may help.
If that doesn't work...the only reasonable course is to replace the caliper.
As far as pushing out the piston, reattach the line and try again. Even with air in the system, it should be enough to push out the pision.