Really tough question to answer given the circmustances-not handy and don't want it to look ugly. It is possible to un-weave chain link but that would require a replacement link. The box stores (Lowe's, Home Depot, etc.) have some but the shortest section is probably 25' and may not be the same height as your dog run. Temporary repairs could be made with wire ( farm supply type stores carry small rolls) which could be used to weave a patch, attempting to duplicate the pattern of the dog run).
You might consider calling a local fencing company and explain your situation. At a minimum they might have some old extras laying around and "home run" would be a sympathetic owner that sends someone out for free. Be sure that you mention your husbands 4 tours of duty-being a 22 year veteran sure pulls at my heart.
If possible a picture of the damage would help.