Whenever the WRI approach gets too convoluted and you get lost in the thickets it can help to retreat to basics.

Here is a definition for Planck's radiation law.

PlanckLaw[\[Nu]_, T_] := (2 h \[Nu]^3)/c^2 1/(E^((h \[Nu])/(k T)) - 1) 

Here is data for the input units:

data = {h -> Quantity[1, "PlanckConstant"], 
  c -> Quantity[1, "SpeedOfLight"], 
  k -> Quantity[1, 
    "BoltzmannConstant"], \[Nu] -> \[Nu] Quantity[None, "Hertz"], 
  T -> T Quantity[None, "Kelvins"]}

and the output units

outputUnits = 
 Quantity[None, 
   "Watts"]/(Quantity[None, "Steradians"] Quantity[None, 
     "Meters"]^2 Quantity[None, "Hertz"])

Then we substitute the input constants and variables into the PlanckLaw and divide out the output units to obtain a pure numeric expression. But there is one catch. Steradians are dimensionless but they are not removed from the expression. This may be the reason the plot was not made. The solution is to remove it by a rule.

step1 = PlanckLaw[\[Nu], T] /outputUnits
step2 = step1 /. data
step3 = step2 // UnitConvert[#, "SIBase"] &
NPlanckLaw[\[Nu]_, T_] = step3 /. Quantity[sr_, "Steradians"] :> sr

The plot can then be made:

Plot[NPlanckLaw[\[Nu], 300], {\[Nu], 0, 10^14},
 PlotRange -> All]

Not working under Windows 10 Professional with Mathematica 10.4.1 too,

PlanckRadiationLaw[Quantity[10000, "Kelvins"], "SpectralPlot"]
PlanckRadiationLaw[Quantity[300, "DegreesCelsius"], "SpectralPlot"]

return empty plots.