The best way to rewrite this sentence, which seems to be a bit droopy in syntax, is:
"The strong belief she had had in the brouhaha she had made, and it's capability to have had ensured the neighbours heard her, gave her enough strength to push the door open."
Of course, this sounds very awkward as well, so here's another alternative:
"Her belief in having successfully alerted the neighbours due to the commotion she made, gave her enough strength to push the door open."
Then again, this not a sentence which feels nice to read out. More context would be beneficial to appropriately rephrase the sentence you've provided.