The ignition switch is located on the left hand side of the frame, just behind the headstock. It pokes out through a hole in the plastic fairing piece that sits over it – this guide starts at the point where you have removed said fairing piece and can see the switch:

There’s a Honda special tool for removing the nut that surrounds it; if you don’t have one of those you can use a drift or punch and a hammer to loosen it (anticlockwise as usual):

Removed:

Now you should be looking at this:

On the flipside there should be a rubber bung covering the opposite side of the switch; on this bike it has been removed and you can see the back of the switch:

Wiggle the switch out backwards through the hole on the right hand side:

On the early 90s CT110 we have four wires coming out of the switch leading to the wiring loom – the CT90s have two from memory but we haven’t got one handy to confirm that with:

These are just bullet connectors and should be able to be easily disconnected. You may wish to check the condition of the connectors while you have them disconnected; make sure they aren’t corroded or loosely fitting.
The new switch, off eBay:

It comes with a small rubber gasket and a thicker nut:

There’s also a rubber cover on the back side that we couldn’t get to fit inside the CT110’s gap – it’s easy enough to remove if you also can’t get it to fit:

The rear of the switch:

Disconnect the old wires, hook up the new ones – matching colours:

Push the ignition switch through from the right side – making sure that you align the frame tab in the hole with the cut-out as you can see here:

Rubber gasket in place:

New nut in place, installed with a drift once it got hand-tight due to lacking the Honda tool or the desire to make a custom socket – note that the new nut on this particular unit doesn’t fit through the hole in the plastic downtube fairing piece:

Original Honda unit on the left, new one on the right:

We ended up taking off the new nut and re-using the old Honda piece so as to be able to use the downtube fairing piece again.
…and you’re done!