MacBook unibody 2010 will not charge reliably
Unfortunately you've tried everything... If there is enough power left to boot you could go to the Apple Menu > About this Mac > System Report > Power > AC Charger Information and have a look if it recognises the charger, and check the battery health.

If this isn't an option because the battery is flat, or health is good and the charger isn't recognised. Then sadly you are going to have to leave the house.