Electrical – Installing smart switches, instructions don’t match the wiring

I'm trying to replace a couple switches with smart switches and the instructions provided with the switch I have (TP-Link HS200) don't match my wiring set up 100%. I'm thinking (hoping) I have the right switches, and that I just need to modify the steps a little bit.

Here's my current set up:

• 2 switches sharing the same wiring box. One controls the ceiling fan while the other one controls the ceiling fan LIGHT as well as an outlet on the wall
• 3 bundles of wires coming into the box from somewhere in the wall: upper left, upper right, bottom middle
• Attached to the fan switch (left): 1 copper ground wire, 1 red wire on the side (bottom), 1 black wire on the side (top)
• Attached to the light/outlet switch (right): 1 copper ground wire, 1 red wire (bottom), and TWO black wires (one on top, and the other wrapped around the bottom screw)
• Wires NOT connected to the existing switches: 3 white (neutral?) wires that are twist-capped together and tucked in at the very back of the box

What I've tried to do:

• Fan switch (left): connect black and red wires to the black "live" wires of the smart switch, connect the ground wires, and connect 1 of the white wires to the white wire leading to the upper left corner
• Light/outlet switch (right): connect 1 black wire to 1 black wire on smart switch, connect the other black AND the red to the other black smart switch wire, connect the ground wires, and connect both remaining white wires (upper right and bottom middle) to the neutral smart switch wire

Results:

• Fan switch doesn't power on at all
• Light/outlet switch is powered on and I was able to set it up on my wifi, BUT it doesn't then the light/outlet on or off at all

Other notes:
The respective switches are the only switches that control the devices, so I don't believe it's a 3-way connection

Does anyone know what I did wrong? Or am I just screwed with my wiring set up and I can't put a smart switch here? Or can I put a smart switch, just not this brand?

This is my first DIY project and my first post on here. Sorry for the long winded post! Thanks in advance for any help!