Hello again Mehrbod,
Thanks for keeping us posted on your findings. 95% is huge! What a disappointment...
If 4) off-target effects is to blame, you can verify that by BLASTing some of your most abundant unidentified reads against the mouse genome on the NCBI website. Or maybe they are from the human genome, and you can search for that too.
At one point, a collaborator was having trouble with their MiSeq and lot's of Phix was ending up in their samples. Illumina spikes in the Phix bacteriophage as a positive control, so maybe these are the source of your reads. You could search your reads against the Phix genome to see if this is happening.
It's a science mystery! 
Colin