I use WLED a lot with strip lights and things and it works great
If you do WLED you should have no need for a bread board. Just get an ESP8266 with male headers and a nice(ish) 12V power supply that can run your strip. From Arduino to LED strip you only need to connect the data pin and ensure 12V PSU ground and 5V PSU ground are connected.
I am leaving the hiding of wires out because I have no clue.
I am curious as to what strip you are planning to use as you will need to (at least from my perspective) hack it apart and extend it with wires to get it nice.
(I advocate for RGB LEDs even though you only need warm white; the stupid stuff you can do with RGB on top of it having warm white is worth it)