Capacitive current spikes will not improve slew rate as much as using a lower driving impedance would ( voltage source) The received signal shows a small step voltage from integrating the spike current.
Since LED's have capacitance at 0V the slew rate depends on the Vol source impedance when using a high side driver with open collector or Totem Pole TTL.
To improve slew rate, you must drive it at the ESR impedance of the LED or lower.
A 65mW LED is between 15 and 20 Ohms ESR
A 0.5W LED is between 1 to 2 Ohms
A 3W LED is around 1/3 Ohm
Thus you need a complementary driver like CMOS with 3V to a white LED or with 5V logic with a small Series resistance to limit current. ALV type CMOS is 25 to 50 Ohms ESR output impedance depending on family , calculating Vol/Iol. Using devices in parallel will reduce the source impedance and drive current so a series R is chosen with ESR of LED and supply voltage. The supply voltage could be used to regulate the current with care or add a small Rs to limit current.
There are other factors which limit the slew rate in LED's beside source impedance and diode capacitance, so high slew rate is a challenge. As ESR drops with rising power, so too diode capacitance increases so this Figure of Merit as the C*ESR product is never spec'd , it varies with different OEM's designs and chip performance.
Low driver inductance is also needed , so keep leads short and avoid loops.
IR LED's tend to be faster and are used in high speed communication like IRDA2. >1Mhz
Look for IRDA 2 drivers , transmitters, receivers and Rx/TX chips.
Remote control IR LED's can be pulsed with 1A at 1.5V but the us size pulses are just run at a carrier frequency between 25~60kHZ then the burst is modulated for data.