What type of material is flexible but not too much, water proof, and can tolerate 80°C temperatures?
The project you linked is for a small heating pad on a chair, and uses insulated wire to prevent electrical shorts.  This is relatively safe (though not as safe as just buying one) mostly because the person using it is awake and alert - if it starts arcing or overheating they will notice.

Building your own to heat the whole bed is a much riskier proposition.  You (and potentially loved ones) will be asleep and may not notice such problems quickly enough.  Using uninsulated wires just makes it that much more dangerous.  If you start a fire, you will probably find your insurance won't pay out after the fire inspector determines the cause to be your jerry rigged heating blanket.

I am all for DIY, especially building my own electronics, but sometimes it is best to leave it to companies that can afford to do a lot of safety testing.  

Heating blankets are easily available at many stores, and probably won't cost much more than the parts and material you need to build your own. They are tested for safety and usually include cutoffs to prevent overheating and shut off if more than the expected current is drawn (if a wire shorts inside).

The linked project says it is more durable than a heated blanket because it doesnt have to be cleaned.  But I have had one for years, washed regularly, with no issues.  If that is a big concern put it under a plastic sheet so it never gets dirty.

So just buy one, and use your heating wire for more interesting projects, like foam sculpting or building an awesome fog machine.