Broken axles on freewheeled rear wheels are common because of the length of unsupported axle on the freehweel side. Even if this wheel seems fine to ride while broken (and they frequently do), it's not a good idea. Go buy 2 new axles. Take the old ones with you to the LBS to get the length right and try screwing the cones on the new axle. Axles come with 2 different threadings, and they are so close that you can turn the cones 3-4 revolutions on the wrong axle before they bind. Buy cones wrenches also if you want to do this job yourself.

Looking at your first picture of the L.H.S. above, I'd say the axle protrudes beyond the outside of the dropout face and is too long, yet in the 2nd picture it looks a bit recessed. The quick release doesn't work right unless the axle is recesses about 2 mm.