The "Cloud" icon is only present for registered users. If I click it it presents a login prompt, and I have no desire to create an account (sorry) so I don't know what lies beyond. It's possible pressing delete deleted something from there. In that case, there would be nothing to recover via Dave's method, as it would have been saved on their server (as suggested by the cloud icon)
If I may present an alternative suggestion; considering you will likely have to restart anyway.
What you are doing is using an online SNES Emulator; the game you are playing is an SNES game, not an online game; the features, saves, etc that surround it are features provided by their emulation.
As you've found, having it in your browser has caused you some trouble.
What I'm getting at is that there are alternatives for Emulation Software. For the Super Nintendo console,
SNES9x is one of the more commonly used ones;
Higan. Both of these allow you to emulate Super Nintendo cartridge games.
The problem is going to be finding the games. the ROM images of the cartridges that are used by emulators- including the online emulator- are copyright material, so providing a link or anything of that sort here would be, quite rightly, against forum rules, but they can be found with a bit of Google. (Though, this entire thread breaks forum rules already since playing SNES games on an online emulator is piracy, though I doubt you were aware of this)
At that point, you'd be dealing with a separate piece of software, rather than your browser. (And, the local emulator software tends to be much better).