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To supply 5mA to a 5kOhms load would require a voltage of 25V.
Do you have this much voltage headroom with you?
You can make a linear regulator and hook it up as a constant current source to get this. You would need supply of atleast 25V + 3V to do this.
The current source output node can see a voltage variation between 25V (5k load) to 2.5V (500 Ohms Load), i.e. a voltage swing of 22.5V. This can mean some power inefficiency with a linear regulator.
With the simple current mirror, the current flow will change with the change in the lad circuit.
If you use some cascoded current mirror, you can minimize the change in current due to the change in load.
Or you can use some sort of feedback to regulate the current.