I do know about your budget, but I would suggest to buy a silent 3d printer like Prusa i3 mk3 which is about $1k. If you decide to tinker your printer instead than take a look on other possibilities:
Most of the noise is created in motors.
Definitely switch to Trinamic stepper motor drivers. I have upgraded from Prusa i3 mk2 to mk3 last weekend and the noise reduction is significant, my estimate is 80-90%.
Take a look on this awesome blog post and video from Thomas Salanderer about the driver. Other good post is on instructables.
I would not suggest soldering unless you are a skilled hacker.
Better option is switch to different board.
Prusa i3 mk3 is using Einsy Rambo 1.1 for $120: https://ultimachine.com/products/einsy-rambo-1-1
But there you have to switch to 24V Power Supply Unit and you have to also switch to 24V heated bed.
Another solution is to use RAMPS and buy trinamic drivers separate (RAMPS doesn't solder drivers on the board).
Additional info about trinamic drivers: https://shop.prusa3d.com/forum/hardware-firmware-and-software-help-f64/tmc2130-driver-infos-and-modifications-t15835.html
Dampers help you reduce noise. There are two types of dampers:
Video about silencing the Prusa i3 MK2 & Horrible Vibrations: https://www.youtube.com/watch?v=OnfYA5QLA84
An enclosure should also help to reduce noise.
How to build enclosure from Ikea table: https://www.prusaprinters.org/cheap-simple-3d-printer-enclosure/