Deriving Planck's Constant from Wien's Displacement Law

So I’m reading an introductory book on Quantum Theory (David Park, 3rd ed.) and I am having trouble with the following question:

“According to Wien’s displacement law, the wavelength $lambda_m$ at which blackbody radiation at temperature T has its maximum intensity is given roughly by $lambda_mT simeq $ 3 mm K. Assuming that the quantum energy at this temperature is of the order of kT where k is Boltzmann’s constant, estimate the value of Planck’s constant.”

My attempt at a solution is as follows:

$lambda_mT simeq$ 3 mm K $,quad$ $lambda = frac c nu$ $quad rightarrowquad$ $frac {cT} nu = 3cdot10^{-3} mK$ $cdot (frac k c )quad rightarrow quad$ $frac {kT} nu = 1.38cdot10^{-34}Js$

$(frac 1 2 mv^2)_{max} = kT_{max} = hnu – ephi$

$frac {kT_{max}} nu + frac {ephi} nu =h$

$1.38cdot10^{-34} + frac {ephi} nu = h$

At this point I get stuck, the orders of magnitude and units seem to be right but I’m not sure what to invoke/what is given in the question that can solve/eliminate this last term to get h, or if I’m even on the right track.