The only reason you would use an inductive Back EMF pulse to drive an LED is called a Boost Regulator.
A boost SMPS basically uses a low side switch ( N PN or Nch) pulsed repetitively at high f and draws a current ramp dI/dt=V/L for the duration of the pulse. THen when turned off, it releases a high voltage in the opposite voltage polarity (rising +) but at the same current and direction for a duration of L/R=T thru a reversed diode, ( thus current is switched from transistor to diode)
The diode current pulse is usually clamped to some voltage or storage cap and in this case the load is a string of higher voltage LEDs.
Thus you have a DC-DC inductive boost or "Flyback" Switched Mode Power Supply or SMPS.
Of course DC would just saturate an Inductor core and current limit the supply or shutdown, since the ramp-up in current is fast.dI/dt=V/L= 12V/100mH= 0.12A/ms. But, Relay coils have resistance that limits the current with fine wire and is always specified for each coil volt for ~ constant power.
Of course for giggles, and amuse your friends, if you have a big old car SPDT relay with the normally closed (NC) contact in series with the coil using a 9V battery or 12V to drive a relay. You now have an HV arcing buzzer. buzzzzz. ouch. Frequency and heat increase with voltage so 5V might last longer but just as intense high voltage.
Often an RF suppression cap across the supply near the relay is used. But for a few minutes of buzzing, depending on the contact gap, you can generate many kV or about 2kV/mm arcs with two wires . Just don't put the current thru your heart between two hands. It may hurt. But very easy corona and you can spread the contacts for more arcs for a few minutes until the contacts burn out.