The blue part is indeed the fill valve.
Start here:
http://www.bayplumbingsupply.com/images/documents/voretolavallefillvalveinstructions-rev06-1.pdf
If you have the Type B valve as shown in the link above, you can service it as shown in the link above, and in the video below.
If you have Type B valve from above, and just cleaning everything doesn't make a difference, try replacing the inside cap with Korky R528. After 9 years, sounds like the rubber diaphram may be wearing out. Look at the link below, and, for example, the rubber in your diaphragm doesn't look nice and flat like the picture but rather scrunched, just replacing the cap for $3.00 might solve the problem. Usually just changing that cap cures a lot of ills.
Here's a picture of the R528 cap and one place to get it:
http://www.homedepot.com/p/Korky-To...ap-R528CM/202666007?keyword=r528#.UbVBwvn5m6g
Regardless of which valve you have (Type A or Type B), if you want to replace the whole valve, get a Korky 528MP valve at Lowe's. Easy to install. For your Drake, you either need the 528T "Fits Toto" valve, which has a blue top, or the 528MP "MaxPerformance" valve, which has a silver top. You can't use the plain-vanilla white-top 528 because it only has a 20-percent refill ratio (the percentage of water coming through the valve that is put down the overflow tube to refill the bowl), which is the old standard. Your Toto Drake has a 40% refill ratio, which the 528T is fixed at and which is the upper end of the 528MP (which has an adjustable ratio, so you just run it all the way open). Even though the 528MP has more bells and whistles, it is more widely-available and is often available a little cheaper than the 528T. Your local ACE or other hardware store probably has both the 528T and the 528MP. If you go to a plumbing supply store, you can also get the Toto TSU99A.X universal replacement fill valve, which is basically a 528MP with some more bells and whistles. But it's probably not worth the extra money to get that one.
528MP at Lowes:
http://www.lowes.com/pd_336988-868-528MPK_0__?productId=3284948&Ntt=korky&pl=1¤tURL=?Ntt=korky&facetInfo=