Not all fish can handle the heat. I have lost a few while trying it, so if you don't think you have fish who can handle the heat, I wouldn't risk it. Not all fish can adjust to temps that high, even when the temp is raised slowly!
Meds are definitely the way to go if you can't do heat/salt. The theory behind the heat/salt is that the ich can't live in temps above 86 degrees when it is in its free swimming stage. The change in salinity may or may not disrupt the cell walls of the ich and may or may not further their demise. I have seen some pretty mixed evidence about it, but I wouldn't bother with the salt if there is no heat. The heat speeds up the life cycle which accelerates treatment, but without that acceleration salt is not going to help you out any with the time frame.
5 weeks might be right, but it depends. Even heat can take varying amounts of time. Just treat until the ich is gone and then a bit after that. Lower the water level to circulate more oxygen in the water. Hopefully all this happens sooner rather than later. I have used ich attack in the past and it was more like 3.5 weeks for me, but it depends. If the fish look pretty bad, I suggest upping to a harsher treatment like Rid Ich (if you can find it) or something similar.
