+1 for talc, but only when it's clean and dry to start with.
The washing up liquid will probably have swelled the card backing, making matters a bit worse. I'd write that one off, cut it off carefully and start again with talc.
You might try spray silicone mould release agent, used in the plastics industry, but baby powder is a lot cheaper.
The trouble with freezing it is that it will get damp immediately it comes out of the freezer, with condensation. If you can avoid that problem it ought to work, as it does actually make the rubber bit smaller, which Talc doesn't.