Here is a link to the Ifixit repair guide for your speaker.
Bose SoundLink Mini Repair
Use the various guides to dis-assemble your speaker so that you can access the PCBs to see if there are any signs of obvious damage to any components, such as burn or scorch marks, stressed components, even obviously missing components.
Be gentle and work carefully when you work on electronics. You do not wish to introduce more problems than what you already have by trying to fix it too quickly. If it isn't going well, slow down and think it through before proceeding. Just take your time.
Look closely around the area where the speaker's power socket connects to the circuit board, as components around there may have been damaged due to the wrong adapter being connected.
If you are not sure take close up pictures of the circuit boards and post them here so that others can see and offer answers as to the most likely repair options.
It may be a scorch mark or just a mark but as it is a power connector unless its connections have 'melted' due to the higher voltage applied, (doubtful), most likely there is further damage in the circuit which after viewing your pictures (very good by the way) is not obvious.
Without a schematic diagram it will be difficult to locate where the actual problem is.
Perhaps the best option is to source a new I/O board.
Searching online using the board number 3574566-0070, gives results at a better price than you were quoted, I think.
Here is a link to just one supplier. As I said it is only shown to give you an idea of the cost, (roughly UK 42 pounds - $US54 inc. shipping -sorry I don't have pounds symbol on my keyboard) If you wish to search for other suppliers, search online using just the number, do not add Bose or Soundlink Mini etc to the search term as this will skew the results and make it difficult to find.
The only thing I saw in the pictures was that in the 2nd picture from the top the charge port looked as if it got hot (or it was a reflection in the camera) whereas in the 3rd picture it was bright and shiny as though it had been cleaned. So I guess that my answer was based on a probable reason rather than on something that I saw.
If you have a DMM (Digital Multi Multimeter -Ohmmeter function) you can connect it across the power input connections and see if you get a reading. Normally you should get a meter reading of some description. It should not be a low resistance reading and it also should not read the same if you reverse the test leads from the meter so that in effect you are taking a reading' the other way'. If you do it means that there is a s/c (short circuit) somewhere. It may be in the port itself, although I highly doubt it. The components connected to the port are more likely to fail with the application of an incorrect voltage supply then the port itself.
If you get no reading at all with the meter leads connected in either direction, then again you have to prove whether the fault lies in the port, (it is possible) or further into the circuit.
As I said earlier without a schematic it will be difficult to find the problem. Not impossible just time consuming trying to work out the circuit and then having to test each individual component, possibly 'out of circuit' to find the fault as there may be more than one component that has failed.