UPDATE
I picked up
1. A 15W soldering iron at Radio Shack. Too big for that pcb...clumsy but usable.
2. .032" rosin core solder. Useless...way too big
3. Desoldering braid. Also useless....way too big.
With a magnifying visor to see the connections, I used the point of a sewing needle (still too big) to pry up the prongs of the chip as I melted the solder.
I was able to tack down the chip into the left over solder on the new board (since I couldn't desolder properly).
I hooked up to a Sabrent Adapter (Tiger Driect) and the drive spun up...no clicks.
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I plugged into the usb of my laptop and after about a minute, the drive popped up! YIPPEE!
I immediately copied all data to the laptop so everything is recovered and safe.
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Thank you so much to everyone here who took the time, with great patience, to guide me through this.
The drive will go back into the tower it came from and I will find a bare bones laptop that can handle audio drivers to record line-in audio. That is the tower's only function. I have an old Acer 4420 Extensa but it is too old and cannot handle the audio drivers with Win 7. That model is no longer supported from Acer so not worth the trouble to try and make it work.
You people really did save the day for me!