I cleared all the pipes under the sink and ran a power snake with my drill through the main line. I didn't seem to hit anything and I didn't get any debris on the coil when I removed it. Any suggestions?

You need to know how long the kitchen drain line is.
Maybe you did not reach the main drain where all the drains connect.

Is the sink draining slow or not draining at all.

If draining slow, us a 50/50 mixture of warm water and vinegar.
About a 1/2 quart each. Let seat overnight and rinse in morning with 2 gallons of boiling water.

If vinegar does not work or totally blocked, u will have to snake from roof vent. Go about 15 feet past sink. Use a power snake that u can rent at a big box store.

Let us know how u make out.

Good luck

Chuck

The first place I would snake is the horizontal line in the wall by removing the trap and running the snake up into the wall. That's only a drain that's 5 ' long.
If you can't get the clog from there the blockage 's farther on down stream. Chuck nailed it with,

u will have to snake from the kitchen roof vent. Go about 20 feet past the base. Use a power snake that u can rent at a big box store.
Good luck, Tom