I am looking for a game that is like Guitar Hero except for PC and for music keyboard instead of guitar. One that I can use my actual MIDI keyboard with. Preferably one that has cool and popular songs like the Guitar Hero/Rock Band series has (and not random weird songs that no-one has ever heard of which rules out any of the Konami Keyboardmania series even if it existed for PC or worked on MAME).

Does what I want exist or should I keep improving my keyboarding skills the old fashioned way by learning how to better read sheet music?

Sounds to me like you're looking for Frederick: Resurrection of Music.
Edit: I can't remember whether you can use an ACTUAL keyboard (i.e. with piano keys) on it or not, though. I imagine it's possible if you remap it somehow...? Not sure about the song choice either.
Re-edit: How could I forget about Synthesia?! Synthesia is exactly what you're looking for.
http://www.synthesiagame.com/

https://en.wikipedia.org/wiki/Miracle_Piano_Teaching_System
Supports PC, NES, SNES, Macintosh, Amiga, Sega Genesis.

If you have a phone/iPhone or (preferably) tablet /iPad, Piano Tiles 2 is probably the most like Guitar Hero for piano. If you want something to sort of teach you to play songs, then Synthesia (suggested by zeogold) is the way to go.

Google gave me this:

Keyboardmania
frets on fire
Music Master Chopin
KORG DS-10
Synthesia
Miracle Piano Teaching System

<span class="bold">Synthesia</span> :  Formerly known as Piano Hero, Synthesia is a video game much like Guitar Hero that utilizes MIDI piano keyboards in conjunction with MIDI files.

  ▪  Developer's website
  ▪  Gameplay video

MIDI interfaces on Amazon, there are a bunch of USB ones.