Thank you dextercioby, my mistake was in assuming that I need to find the unknown function f(λT). I was able to figure out the answer based on your hint.
@TSny, I was wondering if you might be able to give me a small hint in regards to how to proceed with this problem only using the ρ(λ,T)=f(λT)/λ5 form of Wien's law. I tried integration by parts but that just led to a more convoluted expression. I see that you underlined the phrase "product of λ...