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You can’t. If you need a Voltage that is negative to ground then you need some type of inverter circuit. There are a few small chips that can supply a few mAs at -10V. The ILC7660 is one I use but a MAX232 can be used. There are also 555 circuit to produce a negative.
Exactly how are you wiring this? +5V to LED to resistor to MOSFET Drain to MOSFET source to Ground?
What MOSFET are you using??
There are MOSFETs that have Logic level gates that eliminate the need for a negative supply.
Could be that the OP has a P-mosfet, as than needs a negative gate voltage wrt source. So yeah, what mosfet are we talking about?
One way might be use an opto-coupler. It could switch ~5v to the gate of a FET used as a low side switch. (assuming the -10v isn’t really hardwired to the gate as stated in the OP)
Depending on the LED current you might eliminate the FET.