This is pretty tough, it's hard for me to figure out what black would do.
Tactics of the day 11/1

Actually no, theres also a checkmate possibility to,

guys, although almost all the variations are flawed with a cooperative opponent, Kf5 after h4 is indeed leading to mate. but what if Kg4 instead? Mike- you started looking at it, continue that line with less cooperation by black.

take a look at what happens when Kg4 is played right after h4. I agree that Kf5 after h4 eventually leads to mate (not exactly as shown in your variation, but it's hopeless either way).
Try to find lines where black only playes necessary moves and tries to be as active as he can.

I really don't see what else black can do after Qd4+. This is his only other option, which also leads to mate.

This analysis is very good, and indeed it proves that Be4 loses. But Be4 is not the only response for Qd4+, what else could be played by black?
Remember, you looked at one variation (Be4), this doesn't mean anything outside this line regarding other options for black instead Be4. If you saw that one line loses, go to the others until you cover them all (and when only necessary and useful moves are played for both sides, and not cooperative moves with the opponent's plans (for example, why even show the second variation with Rf4?), you significantly narrow down the number of variations you need to look at).
There are 3 styles of playing in the following position- the "algorithm programmer", the "WWE wrestler", and the "Yokozuna". The first two win- the programmer successfully finishes his algorithm quickly and goes home to his loving young and caring wife, no bruises or cuts whatsoever. The wrestler wins his championship belt after a long and exhausting theatrical- artificial fight, to go home all bruised and broken.
Now, the "Yokozuna" loses the fight. Pushing the opponent all over the ring only led to tripping and falling- and the battle is lost. He returns home humiliated, having lost to an opponent smaller than him in size by half.
Which one is you?