Hello,
Since you mentioned a Wi-Fi button, this seems to be the HL-L2370DW.
https://support.brother.com/g/b/producttop.aspx?c=us&lang=en&prod=hll2370dw_us
To begin with, you could try to print a network configuration report (page 75 in the manual below). This will show whether the printer as such is working, and will provide basic network information. Also, have a look at page 171 for troubleshooting steps.
https://download.brother.com/welcome/doc100810/cv_hll2310d_use_oug_c.pdf
Check that AirPrint is enabled (page 132 in the manual). When there is a problem with AirPrint, it may be a good idea to try the following power-cycling sequence as well: Power off the router, the printer, the computer, and all other devices (iPhones, et cetera). Wait two to three minutes. Power on the router. Wait two to three minutes. Power on the printer. Wait a minute or two. Power on the computer. Finally, power on all other devices.
If absolutely necessary, as a temporary workaround, you may want to test a direct USB connection. As VikingOSX mentioned, AirPrint does not work in this case, so you will need drivers. For some other models, a printer driver for macOS Catalina 10.15 has been reported to work even under macOS Big Sur, but that would be experimental. In the specifications for the HL-L2370DW, PCL6 emulation is mentioned, so a generic PCL driver could possibly be an alternative.