i suggest better to go to an electronics repair guy who has better equipment including soldering iron with fine tip...and he will also help you to check the connectivity as he will be having a multi meter handy and understanding about the chips and capacitors installed on PCB.
normally battery mechanics have rough tip soldering irons which they use mostly to connect carbons terminals in stater motors which is useless for this fine job.
13k for battery temp sensor is too much...