I think there would have to be new spren for the new KR. The spren are only as dead as the original KR's vows. If they'd kept their vows until they died then the spren would have unbonded at that time to wait for someone new who is worthy to bond with. As it is, they broke their vows, killing the Spren, then died. The spree are dead without their original knights. At least i hope that's how it is. Too many books have surprise reincarnations and stuff. Can't the occasional massacre just stay a massacre?
As for stormlight reforming spiritual bonds, my assumption is that there has to be something physical to represent that bond, like Kaladins arm, otherwise where is the stormlight to go?