The peak current of a switcher is approx (wait for it) twice the average, so your inductor needs to handle double the input current min, preferable 3x to be safe, if its a one off circuit then go for 3x the max of the 2577 as mentioned 3amps.
Because all the functions are in the chip there isnt a lot to worry about, the only other thing you could do is to limit the circuit max current it can draw from the usb port, but I'd worry about that if and when the problem occurs.
Looking at the photo of your meter I'd say you had an accident when you opened it, where its says 'com' and 'rx' on the bottom of the board near the input sockets there looks to be some screwdriver damage, and there are a pair of surface mount miniature resistors that look like they ahve been ripped off the pcb.
You might get away with soldering them back on the board, then you can look at what actually is wrong.
Or if it cost a few quid call it a lesson learned and buy another, dont worry about looking daft, even the experienced among us do, just that only a few will admit it.
Edit if you put a 0.1 ohm resistor in series with the inductor then put you =r 'scope across the resistor you'll be able to view the current through the inductor, sort of using your circuit as the test bed, be carefull though the 'scope ground might be earthed and so too the ground of the usb port, things might not agree.