The corn syrup helps keep the candied peel supple. Note that Karo and similar corn syrups are not high-fructose corn syrup. You can read more at my post
Why and When To Use, or Not Use, Corn Syrup.) You could use Golden of Agave syrup in its place, or leave it out.
A candy thermometer is very helpful to use here. I have an
old-school thermometer that I've had for decades that I use (as shown in the post) but they can break, so recommend one with a
metal back.
Thermapen probe-type thermometers are great but you have to continue to dip the probe into the syrup while the peel is cooking.
If you don't have a candy thermometer, you can make the peel but you'll have to gauge when it's ready by sight, which isn't as accurate. The peel is done (in step #2) when the syrup is almost all reduced and the syrup becomes white and foamy. But do keep an eye on it if going that route as it's very easy, and emotionally painful, if you go this far with candying the peel, and burn it in the last few seconds.
Storage: I generally keep the peel at room temperature if I plan to serve it within the next few weeks. You can refrigerate it in a container, although over a period of time, depending on your refrigerator, it can get damp (and subject to mold) or too hard. If the latter, you can chop it and rehydrate it in water, and use it in a cake or cookie dough, or fold into ice cream.