Question about Brother Dcp-L2550dn

via WLAN ?! a) from client via WLAN to the printer or b) integrate the printer via WLAN into the network.
For a)
Basically, an existing WLAN from an access point or router is sufficient.
Connect the printer directly to the router using a network cable, for example.
Then just make sure that at least LPD, IPP, AirPrint, Mopria, Web Service, HTTP are active under Network / Protocols (via web interface), so that printing via WLAN with mobile devices as well as WIN/Apple computers should be possible.
For b)
there is no simple solution. The USB port at the back of the device is only used as a print port directly from the host computer.
and the port in the front is used for printing/scanning files.
-> e.g. configure a WLAN access point as a client and then connect to the printer via LAN cable.
Otherwise, switch to another model, as the "N" stands for network-compatible and "W" for wireless-compatible, e.g. the Brother DCP-L2530DW.

I have no experience with a USB-to-WLAN adapter. However, I have connected the network cable (LAN) of the printer to an Apple Airport and integrated the device into the WLAN. This works without any problems.