@Headquarter8302 Pretty sure you're misreading the mention of PNG in the ICO article... the important programs for reading ICO are browsers and the PNG mention was specific to Windows Vista and onwards.
Wow. This is a lot of replies over a favicon. Typically, you don't really need a favicon larger than what headquarter provided but, if you really want it, here is one that is 128x128. It is too bad about how your phone treats ICO files.
Well, you said you would have access to a desktop in 30 days, right? The only other options would be if you find some way to access a more complete file directory (I know Android devices can do this) or you just give one of us temp admin and we can do it for you. However, I don't think a favicon is so important that it can't just wait.