I think they are less looking for blood and more looking for decomp fluids, which with DNA on the mattress from use will show that is her mattress, and that provide a timeline of her estimated death. That would include all of the fluids you are describing.
I am *assuming* we are not dealing with large amounts of blood that would show she bled to death on the mattress or suffered major wounds because they have been in the current residence and the prior and that kind of a crime would leave blood evidence on the walls, ceiling and carpets which there is no way these two cleaned up beyond forensic finding.
However she died I think she was there for awhile and I don't think it was a particularly bloody undertaking.
moo