lost my music/itunes library during restoration from backup

this is probably a simple restoration issue, but i restored my macbook air from a backup and somehow it didn't restore my music library correctly, such that i lost lots of metadata, all my playlists, and a very random collection of album art. i've let this issue sit and kinda fester for some time now, and i've been restoring album art manually; i can't figure out the logic to which albums lost their art. i can't go and check if it was like all albums with titles whose first initial is, say, the letters A through F or something like that. additionally, digging around on my backup disk, it looks like all the metadata (song titles, artists) is missing from every track, from the tracks themselves? as though the music app erases all the metadata and keeps that in the itunes library file? i don't understand time machine at all, how it is working with the music app.

that's the issue described at length. the question then, reduced to one sentence is this: what do i need to restore from backup such that i have my itunes library back, my playlists, and tracks intact with metadata and album art?

(a further issue would be that some albums where one track flows into the next now have a gap of a second or two between them. (sometimes it feels like several seconds of silence were added between tracks) how could i fix that?)

insights and suggestions would be most appreciated.