AC analysis from a DC fresh grad/point of view
What you're missing in the second approach is that \$I\$, \$V_R\$, and \$V_L\$ have unknown phases, even if the magnitude of \$V_R\$ is known. KVL still works, but you have to be consistent in your treatment of complex numbers.

The 25V is an error. The magnitude of \$V_R\$ may be 85V, but its phase is not the same as the 110V input. The phase of the voltage across the resistor will lag that of the input, because the current through the inductor is lagging the voltage across it. As an extreme (non-physical) example, if the phase of \$V_R\$ were 180 degrees from the input voltage, the magnitude of \$V_L\$ would be 195 volts.