Unfortunately DP Review don't have the 100D as an available camera to add into this comparison, but it's a similar story to the Nikon cameras compared to the 1200D. I've included a comparison to the 700D as it's available for less than $600, assuming you are in the US.
See comparison here.
Now, the 700D is a great camera for videography and benefits from the quieter autofocus mechanism available in Canon's STM lenses, and is also an excellent camera overall.
The Nikon D5300 is superior in terms of image quality for still images, and does a good job on video as well. In fact, the Nikon can manage 1920x1080 resolution at 60fps, while the Canon only manages 30fps. It also supports MPEG-4 and H.264 video formats (if that matters to you).
Avoid the 1200D or the D3300 since their microphones are mono, compared to the 700D or D5300's stereo microphone.
Conclusion: Buy the Nikon D5300. I don't know what shops you would consider buying from, but you should be able to find at least a single-lens kit for less than $600. If you can find a package that includes a memory card, filter, tripod and bag for less than $600 then you're laughing.