The kids upstairs are quite boisterous this morning - their tray're stompling on the floor and singing loud songs.ly

They're = they are. Their = possessive pronoun... (ie their bedroom). 'Trampling' really needs an object, like a box or a cushion, you can't trample on just the floor. 'Loud songs' isn't quite right - any song can be sung loudly. Better to just say singing loudly, or you could say 'singing songs loudly' or 'loudly singing songs' (that sounds very awkward though)

I'm under no illusions that this band-aid solution will not solve the problem in the long run.

Normally you'd just say 'under no illusion', although your version isn't technically wrong.
'will not solve the solution in the long run'? I don't think you meant to negate the last part again (I could be wrong). You don't think, that your solution will work in the long run? If this is the case, you don't need 'not', it's already negated by the first part...

The kids upstairs are quite boisterous this morning - theiry're trampling on the floor and singing loud songsly.

their --> they're. they're = they are
I feel like it's already implied that they're singing songs.