Searching a Join List for an index
The implementation of indexJ seems to be broken. In particular:

indexJ i (Append _ left right) 
  | (getSize . size . tag) left &gt;= i = indexJ i left
  | otherwise                        = indexJ i right


indexing with i in the right part is wrong. I'd suggest you to create a function that converts a list into a JoinedList (ideally a blanced one), and then test if indexing into the list produces the same result as indexing into the JoinedList.

Also the implementation of tag is missing, but this is a crucial part of the code, without it it can't be really reviewed.

I'd also suggest to implement other operations on JoinLists, in particular creating a singleton list and concatenation, perhaps also prepending/appending an element to a list.

Finally, you're using a Monoid instance for tagging the tree. This is in general a good idea, but the assignment doesn't say anythong about monoids, sobe sure to understand how the monoid there should be used. You never use the fact that m a monoid in your code. If you're unsure, I'd suggest to forget about monoids at this point and use just the provided Size data type.