An iPhone without lock button function is difficult to trouble shoot and repair.
First step would usually be a simple Reset, which you cannot perform due to lock button failure.
Since you believe the iPhone is working except for the screen being OFF, connect iPhone to iTunes on computer to see if it is recognized and seems to be OK otherwise. If not, try to Restore it with iTunes.
If OK with iTunes, you could wait till battery completely runs down and then power it up with charger. I would open up the iPhone and very carefully disconnect and then reconnect the battery. If this is not done properly you will dislodge the receptacle from the logic board and further destroy the iPhone. When you connect the iPhone back to power see if it starts up OK. If not, consider Restoring iPhone with iTunes on computer.
You may need a new screen and while replacing that consider repairing the lock button flex. I have seen perfect used iPhone 4 going for low prices, around the cost of repair shop prices to fix an iPhone in the condition of yours.