Get yourself armed with a length of wire and a spade terminal on one
end, bare the other and try this.
When the car refuses to start, pop the bonnet and disconnect the small
wire from the starter solenoid, put your replacement wire in it's
place, then make sure that the car is in neutral and turn on the
ignition, touch the bare end of the wire onto the battery pos, if the
car starts the problem lies in the ignition switch, not uncommon.
If you have found the ignition switch to be at fault, there is normally
no need to replace it! You can get around the problem by wiring a
relay into the curcuit, this option is cheap and effective.
Try the wire and see how you get on.