While passive adapters (i.e. HDMI to DVI, mDP to DVI) are ok, active adapters (i.e. DP to HDMI, HDMI to DP) can be problematic. I would recommend that you use HDMI to HDMI (or HDMI to DVI) on the one monitor and mDP to DVI on the other monitor.
In my case (currently using an IOGear DVI+USB KVM - which I dislike and am replacing), I am using HDMI to DVI adapter to connect KY to KVM and mDP to DVI to connect KY directly to second monitor (because my (older) monitors only support DVI). I have also hooked them up vice-versa without issue, BTW
Hope this helps,
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The HDMI port on the NUC to HDMI port on the monitor is a good option, the recommended configuration for multiple displays using this Intel NUC you can see it here:
Since you don’t have displayport on the monitor, I suggest you to follow N. Scott Pearson recommendation mDP to DVI on one monitor and HDMI to HDMI on the other monitor.