Ceiling Fan Not Spinning Normally, Replaced Capacitor Already
Looking at the manual for that fan, there is no "extra box" shown, but it does mention that  

"This fan is remote control adaptable (remote control not included). Please refer to instructions provided with the remote control before proceeding with these wiring instructions." 

So my guess is that your extra box is the remote control option, and the previous owner lost the remote, with the fan stuck on "Low". I would remove it and wire it as it should be without a remote.

Another thing to try is the Fwd / Rev switch. it may be old and/or dirty, try flipping it back and forth a few times.