I would have thought a garage would take about 20 mins to get the door card off, after which it should be easy to spot what has broken. It could be the internal lever, or the door lock, or the cable that links them. Repairing it will then depend on what has broken and how long it will take to get a new part. If the garage wants you to go back when they have the part in, it will save time if they leave the door card off in the interim. That would mean driving around with the door controls missing eg, window switches, door speaker. If it's the handle that's broken the cable could be left attached to the door lock so you could tug on that to open the door from inside.
When the door is repaired and the door card replaced, make sure they fit the waterproof membrane properly, otherwise water can seep into the inside of the car.