Aging a scotch ale in a keg with a stick of oak

I'm just looking for a little advice. I brewed a heavily peated scotch ale recently (9%) that I just kegged. It's about a month and a half old. While it was in the carboy I soaked a stick of medium toasted Hungarian oak in some Old Pultney scotch for about a month. Now that I've kegged the beer I added the stick of oak about a week ago. I've got a couple of questions.

1) How long should I keep the stick in there?
2) Should I shake the keg to get the oak taste mixed well or leave it alone?
3) Did I soak it for too long in the scotch and lose a lot of the oak flavor?

Thanks!

1. I would say 2 months
2. Leave it alone
3. I don't think so. I soak myoak cubes for months before I make my coconut rum Porter
4. I'm no expert though

Great thanks for the response. I'll let it sit for a month or two and try it again. I was thinking the oak wouldn't mix well if I just let it sit there but if it worked for you then I'll give it try.