Whatever caused the screen to crack (it doesn't have to be dropped - pressure on that corner could do it, and pressure ON the corner seems likely if the cracks are radiating out from the corner) could have caused internal damage, which is causing the excessive heat. You might go on using the phone for a few years with no problem except heat (and possibly excessive battery drain - the heat is coming from somewhere and the battery is the only power source in the phone), or the phone could fail (totally - as in even with a fully charged battery it acts as f there's no battery in it) after a week. It all depends on what's damaged and what damage the heat is causing.
Your only choices are use it until it fails (or, if you're lucky, until it's so old that it's worthless) or bring it into a repair shop and get an estimate.