The way to capture fast moving subjects like your daughter is to use a shorter shutter speed. No big surprise there. The way to achieve a faster shutter speed, though is often misunderstood. The camera is only part of the equation. The speed of the lens is the other part.

Before you decide to buy an entirely different camera, I would encourage you to try some things with your Nikon J1 first.

• Raise the ISO you are shooting at. In simplified terms, ISO is a number used to describe the sensitivity to light. As the ISO number is doubled, your shutter speed can be halved and still get the same exposure. So if you need 1/40 sec. at ISO 100, you can shoot at 1/80 sec. at ISO 200, 1/160 sec at ISO 400, and so on. Your Nikon J1 produces fairly good image quality up to ISO 800, and ISO 1600 is still fairly usable for small prints or web sized images.
• Use a wider aperture. The lower f-number selected, the more light is allowed through the lens. Like ISO, a wider aperture allows you to get the same exposure with a faster shutter speed. A lens like the 1 NIKKOR 18.5mm f/1.8 will allow you to use a shutter speed much faster than the 10-30mm F3.5-5.6 VR that probably came with your camera. Setting the aperture at f/1.8 allows a shutter speed less than 1/4 as long as with an aperture of f/4.
• Update to the latest firmware for the Nikon J1. Nikon has released a firmware update that addresses an issue regarding slower shutter speeds being selected by the camera in most automatic shooting modes. Here is a link to the current firmware.

By combining higher ISO and wider aperture, you can significantly shorten the shutter speed needed to properly expose a scene. If you are using ISO 100 and f/4 and the scene requires 1/40 sec to properly expose it, by moving to ISO 800 and f/2, the same scene can be properly exposed at less than 1/1000 sec!

For more, please see these related questions here at Photography SE:
Focus problem vs. motion blur vs. camera shake - how to tell the difference?
How to take photos of children in difficult lighting?
Prime lens or flash: which upgrade will most improve baby photos?
What lens should I buy to get better pictures of a one-year old running around a dimly-lit house?

We also have a good number of other questions here with the [children] tag.