Take the rabbit out of the cage and place it in a safe area. Either have someone to look after the rabbit or place it somewhere it won't escape, like a bathtub. Dispose of the used sawdust or straw.
Inside the spray bottle, mix a couple of capfuls of vinegar in soapy water. Shake the mixture up and spray it onto the cage. Use cloths to break down any stubborn feces or dirt. The vinegar will corrode any dirt, and mixed with the soap, will also sanitize the cage.
Use plain soapy water (no vinegar) to clean the cage again. This will remove the unpleasant vinegar smell, which the rabbit won't like. Thoroughly dry the cage using clean cloths.
Reassemble the cage and add fresh sawdust or straw. Check that the cage is assembled properly and put the rabbit back inside.