I have also used Bourbon but not experimented with rum at all. General principles will surely apply though...
I actually would *not* add the liquor, at least at first -- think of it this way: you are making an oak extract as much as you are infusing the flavor of the liquor into the oak. You can for sure way over-oak a batch by adding the liquor (I've done it...), but it is indeed a good way to add oak flavor. My last little bit of oaked bourbon was added to a test batch of cider, and it was pretty much exactly as if I had added the chips themselves, and you can titrate the flavor pretty easily.
I have found that an ounce of dry chips seems to be a sweet spot for a 5 to 6 gal batch; I soak them in the Bourbon for a couple weeks ahead of time, then strain them out of the liquor, and let them air dry for an hour or so before adding them in a muslin bag. I would be prepared to take samples every week or so and see when you get to where you want flavor wise. I generally go a week or two and get to where I want to be. In the case of my oaked rye pale ale, I am dry hopping at the same time. A lighter flavored mead might oak up in a shorter time period, but it sounds like your mead could be pretty robust, and may easily take 2 weeks of oaking without overdoing it...
If you go a couple of weeks and still doesn't seem rummey or oakey enough for you, you can add little bits of the left over rum to taste.