1- I would not do it in the ocean
Lots of reasons, from the potential to damage an ecosystem to the unknowns that could be catastrophic in so many ways. Of course, cost is another reason.
Doing anything in the ocean --particularly deep-- is incredibly difficult and expensive.
2- I would not use a ten mile long shaft with a drill at the end of it.
I just don't think that this is the way to do it. Again, I am ignorant when it comes to drilling of this sort.
What I would do would be to design a drilling robot that would drill the hole and burrow itself down the hole as it works. To continue with the overly simplistic view, the only thing this robot would need would be power (lots of it).
You select a candidate site and build a large solar array, large enough to power the drill-bot.
A starter hole of the required depth and diameter would be drilled via conventional means. The robot would then be inserted into it. It would expand and clamp itself to the hole to begin drilling deeper. If the broken-up dirt and rocks must be removed either the robo-drill has the capability to collect some amount of it and traverse back to the surface or it comes back and a "cleanup" robot would go down to get the stuff out.
It'll take a long time? Is that a problem? The operation could probably have a good degree of automation.
Probably a horrible idea. Don't know.