OP, your situation basically happened to me, except instead of the car breaking down, the driver mistakenly ended the trip in the middle of the journey. I couldn't walk anywhere so I was forced to request another ride during surge pricing. I got a partial refund and ended up out of pocket for a lot more money than the quote for my original trip.
I had no luck pushing Uber's customer service and just wrote this off as the cost of being a customer with Uber.