Concentration from Absorbance vs Time
The only given concentration is the initial concentration [...]


Unless there is a major misunderstanding on my side, you have all the data you need.

Given the initial concentration $c_0$ and knowing the path length $d$ of your cuvette, you measure the extinction $E$ at this point and use - as you have correctly stated in your question - the Lambert-Beer law to calculate the molar extinction coefficient $\epsilon$.

$$E = \epsilon\cdot c\cdot d$$
$$\epsilon = \frac{E}{c\cdot d}$$

Did I mention units? Units are your friend!


Write the Lambert-Beer law with units! 
Percent (%) is NOT a proper unit for the initial concentration of your protein.



  but I'm supposed to be able to calculate the final concentration and maximum reaction rate


Having determined $\epsilon$, you can now correlate at any point along your reaction the measured extinction with the actual concentration of your sample, including the final concentration.

From the slope of the curve of extinction $E\ $ vs concentration $c$, the rate of your reaction can be obtained. 

If the relation is not a straight line, logarithms are your friend.