From what you're said though, sounds like Mycelium to me! You're almost there man!

[HoLY]

"white growth which is partly `fluffy? and looks on the outside like little `strings?."

If it's pure snow-white, it's almost certainly mycelium. That 'kite-string' like growth is the rhizomorphs going for it, it's very distinctive and unlikely to be anything else. It's also very unlikely that the cake can become contaminated - the rhizomes can usually outcompete any nasties.

Please keep us informed, especially if this is your first time, or your first success. Every first fruiting is a tiny, tiny victory in The War On The War On Drugs.