Re: Decking problem: damage screw and drill bit in the wood.
That certainly sounds like a bit of a pickle @Fatsquare.
Is the broken screw and drillbit both stuck in one piece of decking? If so, I recommend removing the board to gain access to the drill bit and screw. Hopefully, there will be enough protruding to grab the end with pliers once the board is removed. I can't see any chance of removing them without removing the board first. Alternatively, you could shift the position of your new screw slightly and leave the drill bit and screw in situ. You'll then use timber filler to cover the hole from the damaged screw and bit.
You could potentially drill the screw out with a cobalt drill bit. But you can't drill a drill bit easily.