@IPatronius
I will tell you where to look in your time machine backup...however, your time machine backup may not reflect the latest version of your iTunes library file....it may be weeks or months old. Therefore I will show you how to navigate to the iTunes library file on your computer (Mac), that way you can copy and paste it all over the place just in case the main iTunes library file gets corrupted. This instruction set if for Apple Computers only....
For the iTunes library file on your computer go to Finder and Click on your harddrive / solid state drive name > Users > your account username (usually it has a home icon next to it) > Music > iTunes
The two files to copy and paste (but NOT in that folder) are iTunes Library.itl & iTunes Library.xml. Select a different location on your HDD / SSD.
You can place a copy in the root directory of your music folder or any other place that you feel comfortable, just not in the same folder that you found them in.
Now for your iTunes Library file via Time Machine here's where you go. If you have everything in working order I would highly recommend doing a current Time Machine backup. Ideally you should do one, at a minimum every 2-3 weeks, depending on how much stuff you download and how much work you do on your computer. It may need to be more often than that depending on your circumstances.
Finder > External HDD or SSD > Backups.backupDB > Device Name (iMac or MacBook / Pro, etc. - I actually have 2 Mac products so here's where I select which backup root folder to look in) > Latest > Users > Your account name (which should be the same name as your iMac or MacBook / Pro) > Music > iTunes (then look for the same 2 files - iTunes Library.itl & iTunes Library.xml)
Same principle applies....copy and paste those 2 files, not in the directory that you found them in, but just about anywhere else is fine. I stick mine in the root directory of the music folder. Just remember that each time you add music to iTunes to replace the extra copies of those 2 files wherever you saved them. They will NOT automatically update, only the main copy in the iTunes folder will update. Just get in the habit of making more frequent copies of those 2 files in case you don't want to do extra Time Machine Backups all the time.
I hope this helps...