The white dots on your fish are actually Cysts created by the
ICH. When the ich is in cyst mode it's impossible to kill (so I've read). Medicines, and high temperatures actually can only kill the ich when it's swimming around your tank. But don't worry, the ich actually only feeds on your fish for a few days, after that it falls off and can be killed. So medicating actually does two things.
A: prevents more ich from attacking your fish
B: Kills the ich that's on your fish once it detaches.
Also, not ALL medicines will kill your cycle. Some Meds like Mardel Coppersafe are virtually inert. Coppersafe is actually so stable it will stay in your tank for a month. Just replace whatever was taken out by water changes. (it'll kill snails though so watch out if you have one)