I would to not try to measure needles with a ruler. Sizers are not too expensive. This one is under $3: http://www.joann.com/joann/catalog.jsp?CATID=cat2874&PRODID=prd31065 . If you live anywhere near a Hobby Lobby, Joanns, Michaels or AC Moore, those stores should sell sizers of about that price.
Are you using straight needles? If so, the size should be at the ends of the needles, on the heads. Here is a chart about the sizes of knitting needles that should help. http://www.yarnforward.com/needleconv.html
It seems like this is also part of another thread, so forgive me if this has already been addressed. But the pattern gives a gauge of 9 sts for 4", and the finished size is supposed to be 24". That is a fairly loose fitting hat.
Did you do a gauge swatch? Was it 9 sts per 4"? Here's what I would do-- find a way to get a hold of some other sizes of knitting needles and then do a test swatch and figure out which size you need to get that 9 sts = 4" gauge. If making the hat with that gauge is just too loose for you, then go down a needle size or 2 and try again.