If you're talking strictly on efficiency, from what I recall, bringing artifacts to 4 is the best RoI. But you'll also just go insane clicking through that.
Personally, in your shoes, I would probably bring artifacts to 16, since I expect you still have quite a few areas to improve. For me, since most of my champs are already just about as good as they can get, and incremental upgrades are few and far between, I take all artifacts to 8 and sell them if they don't hit speed once, and then to 12 and sell if they don't hit speed a second time.
As for how much silver, you can generally assume taking an artifact to 16 will average at around 1.5-2M, and then divide the total points required accordingly. Recently that's been about 4750, and you get about 330 points for each taken to 16, so you're looking at around 20-25M silver.