How could I have won this endgame?

Truly this does not make sense—unless there are doubled pawns or something, an extra passed pawn (and certainly 2) should be a clear win

Your (biggest) mistake was move 45. Moving the king here does nothing. Your outside most pawn can promote before the king catches it here if you just keep pushing. By moving your king pointlessly you lose that 1 critical move you needed, and the opposing king is close enough to stop it now.

Your king is fine where it is here because the opponents king has no available squares to protect his pawn, so if he pushes you just go right 1 square and take it on the next move if required.

The analysis corroborates this

It should’ve been a draw, your two passed pawns can hold the enemy king and tie it down like your king was to the protected passed pawn

no

oh wait I thought the OP was talking about the position after move 46 where it was a draw whoops

I see, unfortunate to have not calculated that. Not sure why I thought that his king was close enough to the a-pawn.

Yes even here it should've been salvageable… the maneuver of 2 unconnected pawns is just more complex than for 2 connected and I forgot it.

I think your problem was simply that you don't know the quick way to see if you can simply queen your pawn. If 45.a5! you simply queen your pawn and win easily. This is extremely easy to see if you are familiar with the "rule of the square." Google that term and learn that simple technique.

Also, after 49...Ka6 your two pawns can defend themselves. Pushing them just made it easy for Black to pick them off. All of this is in the first couple of chapters of Silman's Complete Endgame Course

move 44, trade pawns instead of advancing your pawn and your a-pawn promotes before the opponent's h-pawn, ez win

even 44. a5 wins after gxh3 you just use your king to stop black's pawn next move and black's king is too far away to catch your a-pawn, but 44. hxg4 is much simpler

move 45 was your last chance, as previous commenters have mentioned 45. a5! is winning

A helpful method to use is to count how many moves a pawn is from promotion and how many moves a king takes to stop a pawn. This helps you see whose pawn promotes faster and whether or not it's possible to stop enemy pawns with your king, or for your opponent to stop your pawns with their king.

This method works for me and often I don't even need to calculate lines, I can just count how long it takes for key ideas from either side to unfold.

That first scenario would've been like the finale game in the movie Searching for Bobby Fischer. But the other people are right that 45. a5! also wins. Honestly, there were just so many chances to win, one would think I was trying to lose.