I know how infuriating it can get and how you start to doubt your own sanity. The best thing is to go back to basics.
1) Locate your actual music files.
If you look on your new computer, by default your music should be stored in my music\itunes\itunes media\ if you cannot see folders in there which contain your music it is fair to assume the media isnt on your computer. I tend to suspect that the music is on your external hard drive which is why it plays when you have that plugged in.
If you find the music on your hard drive I would go into preferences in itunes and tick the option in advanced that I mentioned previously, then, in itunes go to file and "add folder to library". If you cannot see the menu bar which contains "file - edit - view - controls - store - help" you may need to press ctrl B to show it, alternatively if you click on the little box in the top left corner of the itunes screen and drop the menu and select "show menu bar"
So now when you press the add folder to library simply browse to where your actual music files are, find the folder which has all you r music etc in it and then click "select folder" in the bottom right of the window.
If you have done all that correctly and have ticked the "Copy media to itunes folder when adding library" option in "edit / preferences / advanced" your music should be copied from your external hard drive into the itunes folder and all should then work properly.
Fingers crossed lets hope thiat works for you.
Just a thought - if that creates duplicate entries in itunes then I would select all your music in itunes and delete it, then go through the above again so you only have the one set of music.
Steve