I think you need to be a little more precise/accurate with your measurement. Look at the Sheldon Brown site and you'll find a chart which explains the dimensions of various headset options. None of these is close to 1 1/4".

Probably if you measure carefully you will find that the inside diameter of the head tube is about 34.0 mm which is the diameter of the cups for a 1 1/8" headset. But then, a standard 1" headset has a cup diameter of 30.2mm, which is closer to 1 1/4", so maybe that's what you have.

Anyway, in all probability , you need either a 1" fork and headset or a 1 1/8" fork and headset. The 1" will measure about 30.2mm and the 1 1/8" will measure about 34mm. That is not a subtle difference so it shouldn't be hard to measure and decide. If you find it to be 34mm, then you are pretty much limited to threadless 1 1/8" headsets and forks. If it's about 30.2mm, then you can go either threadless or threaded. Most folks would probably go threadless but it depends on what you like. Either would work just fine. You might consider taking the frame to an LBS and buying an appropriate headset and having them install it. Many LBSs will install a purchased headset for free -- of course, it's not really free. Then you know what fork to buy and can either have the crown race installed or do it yourself.

Just for reference, these days accurate vernier style calipers with a digital readout are available for not much money at all. Even cheap ones will allow you to measure to .1 to .2 mm accuracy. If you do much bike maintenance at all, you will find them very handy very frequently.