having the hole wander is a big deal
center drill
stub drill
jobber drill
taper drill
then whatever else you have
all new bits or freshley ground on a proper drill sharpening rig
This can be done on a home lathe but a lot of care is required.
Believe it, the hole WILL wander.
To try and eliminate ANY wandering, use NEW DRILLS, that's stub, jobber and then an Extension Drill of the appropriate size. These can be bought as an over the counter item as there are a lot of standard and non-standard sizes available.
You need NEW drills to start the hole to try and eliminate any chance of starting the hole off centre or wandering, remember that a hand held home sharpened drill will create/cause this wandering, I don't care how well you think you can sharpen them by hand, it's not good enough for this application.
If you want it to be smack on centre at the 13" deep end, then you are going to have to go about it with a bit of boring involved.
If there is a length of DRILL ROD available over the counter, that is the exact finished size to the hole you want to drill, then use the drill rod to make a simple gun drill out of it. Start the hole with the smaller sized drills and then sneak up to about 1/32nd" under the finished size and then use the gun drill you have just made as a boring bar/ reamer.
This will work like a single point cutter and bore the hole true. There has been quite a lot posted about making gun drills and hardening the cutting end, so no need to go there.
Stainless steel eh, have you been warned about this stuff work hardening, you get the working point very, very hot or red, games over, you just made that bit of stainless go rock hard up at that point, chuck it and start again, got to keep liquid on it , easy speed and slow cutting feed is the way to go.
Have fun and hope you get the job done.
regards greenie.