How to deal with this water damage
Chances are, now that the leak has been repaired, the moisture will dry up and you won't have further problems. You may have some subfloor to replace when/if you ever replace the finished floor, but it's probably not an urgent matter. 

Since you have an overhanging faceframe you can add a veneer to the side panel of your cabinet without showing an edge at the front. I'd visit your local home improvement store and see what they stock in the way of paneling or pre-finished plywood. You could also buy raw birch plywood and finish it to match. 

You'd then carefully cut a panel to fit the shape of your cabinet and use panel adhesive or contact cement to adhere it to the side of the cabinet. Sand or scrape any bumps from the existing surface first.