Although not a part of the answer, but that is not JSON. That is just a plain JavaScript object. It is funny that JSON document and a JavaScript object are quite the same thing (but, still, naming them correctly can help you get valid and useful answers).
Now for the answer part, you can just change the code where you are generating the option elements and use the radio buttons instead,
$.each(a.Sheets, function (key, value)
{
$("#sheet").append("<input type='radio' name='" + value.sheetname + "' /> " + value.sheetID + " <br />");
});
Something like this will help you out. As for other styling or DOM purposes, add more elements such as the
<label>
etc.