Using gate driver MIC4427 with 24V supply
This driver is meant to switch FETs fast (less than 25ns according to datasheet) which is what you want for DC-DC converters running at high frequency. However your use case is different, if you use PWM on LEDs it won't be 500kHz, more likely it will be at most 25kHz if you want to avoid audible frequency, or 5kHz if you just want to avoid any flicker.
At such lower frequency, switching very fast is not necessary to minimize switching losses... and it is actually a problem because this will send high dv/dt and di/dt into the wires and the LED strip which will then radiate this as EMI. In other words, if you make your FET switch at 10-20ns and you're not sticking an inductor right there to make a buck converter but instead connect that to wires without any filter, you're really making a wideband radio jammer with the LED strip acting as antenna.
Since the FET has a gate charge of 17nC, with say 4mA gate drive it will switch in 4µs and with 20mA it will switch in 1µs, which is both fast enough to produce negligible switching losses at a PWM frequency suitable for LEDs, and slow enough to radiate much less EMI.
Your MOSFET is a good choice for 5V gate drive since it specifies about 10 mOhms RdsON at Vgs=4.5V.
If you use a 5V microcontroller, you can drive the FET gate directly from it with a resistor, say 220 ohms, that should switch in ~1-2µs.
If you use a 3V3 microcontroller, you can use your MIC4427 powered from 5-10V to drive the FET, but you'll need to add a gate resistor to slow it down. That's a bit of a waste of a FET driver, but if you already have it, why not... A cheaper option, especially if you have several channels, is to just use 74HCT or 74ACT logic gate as voltage level translator from 3V3 to 5V, powered from a 5V supply that you probably already have.
If you don't have the necessary supply voltage, just use a linear regulator from +24V. You can't power that driver chip from 24V.
Unless you intend to run a huge number of amps through it you don't need to increase Vgs to more than 5V to scrape off the last bit of RdsON. So if you already have 5V somewhere it'll be fine.
PS: is you need to convert 24V down to 5V consider one of these instead of an old 7805.