The confusion may be that drizzle can be a noun or a verb. For example there is a light drizzle outside - noun. There is light drizzling outside - verb. I am only guessing that this may be a problem that needs explaining.
A eu means had, has had, have had, which is it? In this sentence why can't it just mean, we had drizzle all day or you can say, we have had drizzling all day. But, it has been drizzling all day, sounds better.