2. Remove the broken screw from the chainstay bridge so I can re-fit the mudguard screw.
Ok, this is going to be a bit of a how-to with Easy Outs and what to do when they don't work (because it didn't).
First, if you can, file the end of what's left flat:
Make your life a bit easier - put what you're working on in a position where it's stable and where you have decent access:
Next centre punch the centre of the screw as close to the centre as you can get it. A little tip to help free things off is if you get a little hammer and tap away at the stuck bit for a few minutes. Don't really hit it, just tappity tap tap away. You're not aiming to deform anything, just start to break the hold that the corrosion has. Then drill a hole down the broken bit. I started with 3mm for my Easy Out and chickened out and finished it off with a 2.5mm. You can see I didn't get it perfectly centred:
Insert the Easy Out and tap it in lightly. Again, you don't want to really hammer it in, you just want to get it to bite in a bit so it can get a good grip:
Then you can start to wind it out using a tap wrench. Now, you either get lucky with these or you don't. There is a bit of skill involved, but it's mostly making a judgment call about whether it's about to break, slip, or if it's chewing up. In my case, it didn't budge in the slightest and then broke through the side where I didn't drill the hole particularly well in the centre:
That's basically not coming out. I'd suggest this happened because of a combination of my hole not being very well centred and I could have probably done with drilling a smaller hole. All is not lost though, because my backup plan was to simply drill it out and re-tap the thread. So I went through with a 4mm drill and tapped it with an M5 tap:
You'd normally drill to 4.2mm for M5, but I wanted to leave plenty of contingency metal in there and let the tap do the work instead. After tidying up the outside, I have a nice clean thread to pop a fresh screw into: