The seventh character of the VIN tells you which transmission the 2004 TL has:

5=6-speed manual
6=5-speed automatic

The eighth character breaks that down further:

2=5-speed automatic
5=6-speed manual
6=6-speed manual with high-performance summer tires

The first four characters of the transmission number:

DEC6=6-speed manual
BDGA=5-speed automatic