Why is my air stop working at my battery died on my 2010 Ford Taurus

Hello, first -- see to the resolution of the battery dying concern. Have the battery professionally tested to see if it is fully charged and able to sustain a load. If it's not - replace it. Either way, ensure a known good battery is installed into then vehicle and perform a charging system test. Make sure the battery and charging system are working correctly. Once this is resolved, then recheck air conditioning operation. If it's still not blowing cold -- then this system requires testing. If it's that the air is cold but the fan doesn't turn on, that's another area that would require testing - the blower operation. Use this link to find some shops near you that can help with your concern: www.repairpal.com/repair-shops