The lugs on the speedo drive fit into slots, if it's just randomly put back on the lugs snap off, and were probably the cause of that squeak.
I'd imagine it's a common mistake to make, i did it myself when i first took my wheel off years ago, and i read about it quite often.
But yes it should be packed with grease.
Washing out grease is a very bad idea. When things go round they generally need lubrication.
I very much doubt gluing the lug back on will work either, it'll just happen again probably, and you were fortunate to pass the MOT because i doubt that your speedo is working properly as it is??
I'd get it sorted sooner rather than later otherwise your next thing will be a speeding ticket!
