The resistor values for the level shifter are not critical as long as long as they are approximately in the ratio of 2:1. So 1K with 470 to 510 ohms should be fine. But they shouldn't be much smaller (due to drive considerations) and if too large, there may be problems with the frequency response. Under some undetermined conditions, a 100 pF capacitor between D11 and GND is needed to prevent erratic behavior. There is probably a race condition in the ST7789 library SPI timing.
If you have an mLMA1 PCB, it can be used to test the LCD before fully populating it. The level shifter resistors are all in one block under the LCD. R7-R10 are ~500 ohm; R11-R14 are ~1K ohm. (The specific values included in the kit may vary slightly but should be in the approximate 2:1 ratio.) Or the circuit, above, can be built on a prototyping board or solderless breadboard. Double check the wiring and make sure the connections are secure.
The following are the steps to do a clean installation of the Arduino IDE and device drivers. If there already is an Arduino IDE installed and it is totally screwed up, it should be replaced or updated. This can be done without losing any of your Sketches. If it works, don't fix it unless it becomes clearly broken with the Nano. ;-) But if it's very old, the updating is recommenedd so that most of the old bugs have been removed and only new bugs remain. ;-)
Go to Arduino Software and install the current version of the Arduino IDE (V1.8.13 as of Jan 2021). (I'm not sure if the board needs to be plugged in to a USB port during this process, but mine was. During the install process, it will ask to install the drivers. Reply "Yes" to all its requests. When the Arduino IDE is started for the first time, go to "Tools", "Board", and select "Arduino Nano". If the Nano is plugged in, an new COM port should appear (in addition to any that were there originally) under "Tools", "Port".
If a new COM port DOESN'T appear when the Nano is plugged in, it probably means your Nano is a clone (most are) and the next step will be required.
More info on software, drivers, and more at Getting Started with Arduino and Genuino on Windows.
Several Libraries are needed by the LCD test sketch. Use these links to download the .zip files into any convenient directory. Then the libraries can be installed from within the Arduino IDE. Go to "Sketch", "Include Library", "Add Zip Library".
The SPI library is also needed, but that can be loaded from within the Arduino IDE. Go to "Sketch", "Include Library", "SPI".
A very few of the commodity Nano boards have the "New Bootloader", so if it fails to upload, try changing to the other one: Atmega328P.
ST7789_TST3 is a variation of a number of similar sketches found on the Web that can exercise the LCD without requiring any hardware much beyond the Nano and a few resistors. The normal mLMA1 firmware may also be used. Either of these can also be used to determine if a specific LCD comes up in normal or 8 color mode (see the mLMA1 manual for details.
Sketch uses 18834 bytes (61%) of program storage space. Maximum is 30720 bytes. Global variables use 873 bytes (42%) of dynamic memory, leaving 1175 bytes for local variables. Maximum is 2048 bytes.
It is possible to turn on "Verbose Mode" for both compilation and uploading, but what is displayed ends up being only most useful to other machines. :)
The firmware is retained in non-volatile memory so uploading only needs to be done once - or until a new version is available!
The firmware may also be compiled without uploading by using Ctrl-R. Since you haven't messed with the code, it should compile without errors. This is slightly faster for testing and doesn't use the board at all so it can be off doing whatever it pleases. :)
The LCD_Test sketch displays a variety of text and graphics in a continuous loop which confirms that all of the above steps were successful. Congratulations, this probably only took 10 times as long as you were hoping. :)
If LCD_Test seems to upload correctly but the LCD doesn't do anything reasonble, simply try one of the example sketches like "Blink" to confirm that the Arduino IDE and USB are working. Go to "Files","Examples","01.Basics","Blink" to load it. Then follow the instructions above to compile and upload. Changing the values of the delay(1000); instructions can be done to quickly confirm that code is working.