How do I fix the touchscreen?

If the machine turns on with an external display (AND isolated to the LCD), the machine is probably fine, but the LCD assembly is damaged, or you blew the LCD fuse :-(. Get a DMM and check for a fuse around the iDP connector—it's usually labeled with an F at the beginning. Measure it in continuity, if your meter beeps, the fuse is good, but the LCD is blown.

Sadly, the issue with Chromebooks (and why I disdain them, especially in schools) is that the cost of parts vs. replacement is always at least 50-60% of the cost of a new one - repair is possible, but they are usually BER way more often than not. Sometimes you get lucky and find an MDM-locked donor with what you need, but a lot of those come from schools and tend to be trashed to such an extent that they're awful donor machines since the schools charge the student for the Chromebook, replace it, and recycle it with the MDM because they ALWAYS forget to remove it. Not all of them are worth more dead, but it's like a 70/30 dud to working unit ratio. I wouldn't buy them to refurbish given the high MDM remains rate, but as donors? As long as they're not run over and glued several ways to Sunday, I'll buy them to do shell swaps or pull what I need to fix the other unit - as long as the AUE policy is still active for 3-4 years.