Running Ethernet between floor and finished celing
It may be possible to use the coax cable to pull your CAT 6.
Remove the coax wall plate and pull the cable off the back of the plate (or otherwise get the cable freed from the wall).
Securely tape sturdy, but thin string or twine to the end of the coax. Pull the coax back through the wall to an access point in the laundry room. Doing so will pull the string through the wall, leaving a &quot;bread crumb trail&quot; back through any holes that were drilled for the coax.
Securely tape your CAT 6to the coax cable. Using the tape, make a tapered lead end around the coax and around the CAT 6. I'd suggest that not having an end on the CAT 6 would be the way to go - it's a big square thing that will catch, while the cable itself is fairly small and will slip through holes easier.
Carefully pull the string back through the walls until the coax and CAT 6 pop out at the original location.
Terminate the CAT 6, reattach the coax to the plate, and presto! (You may want to upgrade to a plate that has a coax and RJ45 jack so you have a nice, tidy finished look.)
NOTE: all this is predicated on:

The coax not being attached to the walls anywhere.

Any single wire staple anywhere in the walls will prevent the coax from pulling out in the first place.


There being no sharp bends in the coax cable run.

When the cable was first run with the wall open, it could make some fairly sharp bends. Once the wall is covered and you're tugging from 15+ feet away, it's far less likely.


The cable(s) and string not separating.

You'd probably be better off tying the string behind the barrel connector of the coax, then taping over it as a 2nd method of attachment


The holes in the studs being big enough to pull the coax and CAT 6 through.

It's entirely possible that the original installer drilled a 1/4&quot; hole to pull the 3/16&quot; thick coax through (actual dimensions may vary, numbers quoted for demonstrative purposes only) and terminated the cable once it was run. If this is the case, the coax won't pull back through the wall.
It's possible there's enough room to get the barrel connector through the wall, but not the connector, cable and an extra piece of CAT 6.



If this doesn't work, you may have to bit the bullet and cut a couple of access holes in the ceiling to pull the cable, then patch &amp; paint when you're done.