How to deal with backyard flooding?
Given the constraints, I'd remove the walkway stones, lower either side slightly (and probably bury a perforated pipe, or several to connect them and speed drainage) raise the walkway (possibly just using soil removed from either side), and reset the stones.

Beware of buried utilities - The official ones 811 (or your local call before you dig option if it's not 811) will mark for you, and the ones they won't know about such as wiring between your house and garage. They should be below the extent I'd expect to dig for this (less than a foot) but always check, and aways be wary (smooth black roots are not roots!)