Deteriorated downspout replacement
Personally, I'd cut the downspout back so it is even (level with the ground), with about an inch of pipe hanging down (or as much as can be saved). Get another piece of pipe, in which the inside diameter is the same as the outside diameter of the original downspout. This will be used to replace the piece you just cut off. Before you affix your new pipe to the old, you'll need to put something on what is left of old pipe to save it from deteriorating further. I'd use a rust converter, something like POR-15. Use it liberally on the inside and the outside of the pipe ... wherever you see deterioration. If the new pipe is a snug fit on the old pipe, you can actually use the POR-15 to bond the two pieces of metal together. You'll also want to put some type of screen up top to prevent the debris from getting stuck down in the pipe again, or you'll run into the same issue. You'll need to get some type of reducer which will allow you to transition from the new piece of pipe to the crimped piping which is there now.

This would be the "easy" fix, if you will. If the deterioration is too bad, you'll need to consider a complete replacement of the gutter system. This would cost some serious money, but would bet you are going to have to do it sooner or later anyway. Check any other downspout to ensure the same thing isn't happening there. Use the POR-15 to prolong your pocketbook agony for as long as possible.