What is the necessary action to take regarding a neighbor whose dogs cause severe disturbance preventing sleep, and who refuses to respond to repeated complaints, given the impossibility of resorting to higher authorities?

Your neighbor is not permitted to keep dogs if they cause you harm, because harming a neighbor is a major sin. He must remove the harm, as the Prophet, peace and blessings be upon him, said: "There should be no harm nor reciprocating harm." It is best for you to be patient and advise your neighbor. If he does not respond, then you are allowed to complain to the authorities. If the authorities do not remove the harm, then seek Allah's help and supplicate to Him.