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I am trying to find the maximum current the DIO5661CD6 can supply to the LED(s.) I do not see the usual "continuous current" parameter anywhere in the datasheet.

Is the "Peak NMOS current limit" the one I need?

From page 4 of the datasheet:

peak Nmos current limit datasheet screenshot

If yes, why is it labeled "NMOS current"?

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The peak NMOS current is the maximum peak current of the internal N-MOSFET as shown on the datasheet extract below.

enter image description here

Because it is a switched mode power supply. The peak current into the MOSFET is not equal to the mean, smoothed, current that results from the inductor and capacitor filtering.

The continuous current capabilities of the system is not a fixed parameter that you can find on a table. It is mainly driven by the thermal behavior of the system. You can compute the losses into the different elements of the circuit (mosfet, diode, ...) as long as the resulting temperature is kept below the maximum allowed junction temperature given by the part datasheets, you are good to go.

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