Unless you remove the old spindle you wont be able to fit the new handle as you have discovered.Try drilling a hole in the spindle and then drive a strong screw into it.Now use a claw hammer to lever it out.
Now fit the new handle,the spindle may need cutting to the right length before fitting.
The handle has broken off whilst I was trying to open my window, it's still shut. I have got a screw out of the bit left on the window (bracket?) but I think there is another one under a small plate, how do I get to it? I had the handle missing on another window when I first moved in and I swear the landlord took 2 minutes to replace the unit, it seemed really easy, what am I doing wrong?
the cover plate is held on with a low tack adhesive. Pop the point of a sharp knife in the small gap between the cover and the body of the handle and gently lever it out. Do not worry about damaging it as all new handles come with new cover plates