Can someone explain the nuance of yoku to me?

I've seen よくa lot in my reading lately. I'm currently reading a story at the moment about a ruler of Russia who got sick and requested the help of famous doctors around the world.

よく has come up a lot in terms of saying he will/ won't get better. The doctor says to him :
"王様、この薬を飲んでください。すぐに、よくなりますよ。"

Which I'm taking as "lord, drink this medicine and you'll feel better."

The next line it says he took the medicine but :
病気はよくなりません。

So he took the medicine "but the illness didn't get better"?

Then an old man comes to look at him and it's used here like :
おじさんは、王様の顔をよく見て言いました

Which I translated to "the old man looked at the lords face closely and said"

I thought よくmeant often? Or frequently? Can some one explain this to me please?