The wooden wok cleaning whisk is the absolutely essential tool for cleaning a wok under all normal circumstances. The wok cleaning brush is a very very simple wooden cleaning whisk made of many thin slats of wood which collectively scour out the wok without removing the wok seasoning in the process.
The wok cleaning whisks are available in a few sizes - an 8 to 10 inch cleaning whisk should be fine. Follow directions for normal wok cleaning when you use the cleaning whisk.