Flex bit and placement tool is what you need to drill down into the wall. You're going to have a huge mess if you break that tile floor.

Although, I would cut a couple of feet off of this linked flex bit shaft to make it more manageable to work with. Its the first hit I got in google. My bit is about 3 ft.

https://www.lowes.com/pd/Greenlee-3-16-in-Round-54-in-Drill-Bit-Extension/5001762199

Cut your old work low voltage bracket hole in the wall and install the bracket, then use this kit to drill down. The placement tool easily bends the bit to hit the middle of the sill plate.

I've fished hundreds of low voltage outlets in my career and currently train technicians to do this every month. Trust me, this is the way to fish an outlet down to a basement or crawl.

And as a side note, the person that warned about running comms in close proximity and parallel to power is absolutely right. That can possibly cause some induction issues, aside from accidentally getting into your power insulation with your tools, and should be avoided. You want to keep some space between them.