I've got a Yammy arranger keyboard. It has a MIC input for recording. I also have a RODE NT1 old gray version (not the NT1a), a Behringer PS400 phantom power supply, and an XLR to 1/4" (6,3mm) TS cable, so I hooked this all up last night into the 'MIC/LINE' input of the keyboard. The selection switch for 'MIC/LINE' I left in the MIC position. On this keyboard the MIC input goes pretty much straight away into the first stage of gain (NE5532 OpAmp) with some filtering, but no means to attenuate the signal. The output of the first stage goes to a 'Volume' control prior to going to the second stage of gain (another NE5532). The RODE NT1 is about 20dB hotter than a dynamic mic I would normally use into the keyboard. The RODE NT1a is about 3dB hotter on the output than my NT1. The keyboard has some effects and other processing which I left off, including a noise gate.
Listening with headphones, I spoke at different distances into the NT1. Close up the proximity effect was noticeable and it appeared that the audio was slightly distorted which I would believe was the first gain stage being overloaded by the hot signal. I backed off the mic to about 12" and the distortion subsided, however the room sounds were more evident. I didn't try any recording and the gain setting ('Volume' control) might be able to be set lower to reduce the room sound and keep a good recording level.
In summary, if the OP's DGX660 has a similarly designed MIC input as my Yammy keyboard has, he/she 'might' be able to use the NT1a with the PS400 phantom power if careful about distance to the mic and loudness of singing, otherwise the recording may result in some distortion. An option might be an inline XLR attentuator to knock some of the signal down going into the keyboard, but that's another added expense.
The idea of a mixer could work into the DGX660 'AUX IN', but then the keyboards vocal effects such as reverb are not available using that as an input. Also the mixer purchased needs to be capable of 24-48vdc for phantom for the NT1a, which some budget mixers may only provide 15-18v.
My keyboard and the DGX660 records to a stereo WAV file, so what you record is what you get and no way to mix/adjust the vocal in relationship to the music. The mic inputs of the Yammy keyboards were designed for use with dynamic mics which have a lower sensitivity. I'm inclined to suggest a decent dynamic mic for creating 'live' recordings with the DGX660's vocal effects available and keep the NT1a with the thought of getting a USB interface in the future that can record and track four inputs (one for the NT1a and two for the stereo of the DGX660) into a DAW to create a more polished recording.
I recently got an AKG D5 which runs about $100 and has a crisper output than an SM57..
AKG D5 Supercardioid Dynamic Handheld Vocal Microphone | Sweetwater
EV's Cobalt C09 is also pretty decent for the money..
Electro-Voice Co9 Cobalt Premium Vocal Microphone | Guitar Center