I wouldn't bother with removing everything from your tank. Ick will remain regardless of what you do. As previously mentioned it is an opportunist, and when your fish are under stress they are more likely to be susceptible to it. One outbreak of ick in my tank occurred after an extensive gravel clean, the fish were obviously stressed by it!!!
It normally lies dormant as cysts in your tank, increasing the heat increases the hatch cycle, and so long as your fish are not stressed it is unable to attach to the fish. The ick needs to have a host otherwise it will die, from memory death occurs around the 24hr mark.
White spot cure will do the trick. Coming from a medical background I would insist that you use the recommended dose on the container. These meds have to be tested prior to being sold, and unless they state otherwise they should be safe at the recommended dose.
Otherwise salt is the other option. Tonic will do it, but that tends to stain the silicone. I have found that the whitespot cure doesn't stain the silicone as much.