Other than forcing the Klingons to the bargaining table with the Federation, how did the Empire recover from this disaster? Posting this as a question, but the thread may turn into something more.
that plot thread was dropped after the movie in which it was first mentioned.
BUT, there were a few clues in the movie itself. Short version: the homeworld now needed to be terraformed to undo the damage done by the explosion. If the Klingons were to attempt this themselves, they would fail to get it done in time to prevent the planet from becoming unlivable.
How would this destroy the empire? Well it seems unlikely that more than half the Klingon population lived on Qo'nos. The most likely explanation is that it'd indirectly cause a civil war. That compounded with the loss of both Praxis(their "primary energy production facility") and the HW would leave the empire unable to properly protect itself from outside threats.