OP, don't know about the landlord situation but I think you can and should call Fairfax County Public Works -- Sewage backup number.
I don't think "A little bit of chlorox" is a fix for this situation and neither should the handyman. At least the Public Works people should be able to tell you what needs to be done to fix the problem.
When a friend of mine had this happen in MoCo she called someone from the water dept I think and they sent a cleanup crew -- she said they came in hazmat suits. it was beyond disgusting, what her basement looked like.
Here's a story of someone in Fairfax County:
http://www.angryox.com/sewer.backup/