Check My Geek Math
(TOS / AOS era) Warp 8 = 512 x the speed of light**
1 light year would take a year to travel (at the speed of light) ***
So, this is my math:
So, by my calculations, at warp 8. it would take about 11 days (more, actually, but I don't know how much 'a little over' is) to get to Vulcan from Earth.
Over 16 fucking years at sub-light speed, which Enterprise would be traveling after ejecting the warp core.
I'm thinking they got a tow home. You agree? Of course, maybe they found a base on the way. Yeah, I just now thought of that. I R Smrt!
* info from memory-alpha.org
** info from star-fleet.com
*** It's all in the name.