It means a full quarter of an hour. Besides that, 足足 also have a meaning to describe the duration of the time is very long for the person at speaking. It's like an emphasize. In this sentence, it means the 15 minutes is pretty long for the person who writes this sentence.
足足有一刻钟 here means "a full quarter of an hour".
足足 means fully. The word is usually emotional, implying that that amount is a big deal. E.g. 我在这里足足等了你两个小时. (implying: two hours is really long time. It's a complaining about the fact that you kept me waiting for that long.)
PS. Compare 足足有两小时 and 足有两个小时. They mean the same(a full two hours), but the former ...