Is the blower motor relay also in the photo attached to the BMR? This is that old cowl water leak issue again -- it's possible that water will also enter the high speed relay case and corrode the contacts so that none of the settings will work.
In my '01 LS, the persistent water leak first killed speeds 1-4, then eventually 5 -- if I apply power directly to the blower motor feed it works, but the whole BMR appears to be a dead dog.
I gave the dealer one shot at fixing the leak (they did the case drain cleaning thing) -- fail -- since they also changed the BMR at that time, the new BMR quickly died. Since then, the windshield has been replaced for a different reason ($219@SafeLite) and the leak appears to have been fixed, but it hasn't rained enough since the repair to claim success yet.
Before I apply the repair to the BMR myself, I may apply some mods to the blower case while I'm in there to make sure that any water stays away from the BMR -- maybe drilling some drain holes in the case to direct the water away from the BMR or at least prevent a puddle from forming in the case.
Winter is here and I need heat!