Oh wow! As much as I love all the info for DSLRs, and will further look into, I was specifically referring to iPhone photography
Sorry I wasn't clear!
Do you have any links for the "natural light" flashlights? My guess is that they're expensive?
LoL Leanna, I kind of suspected you might be referring solely to iPhone photography, but since I know you're also a DSLR shooter, I thought I'd spread around some more info.
As far as "natural light" flashlights, I guess the more proper term you'll want to look for is "High CRI" flashlights. CRI stands for "Color Rendering Index", and is a measurement of how well a light source can produce the best color when compared to "natural" or otherwise ideal lighting situations. Basically, you want to look for flashlights that have a high CRI. The only problem is that not many flashlight makers actually advertise this. There are, however, plenty of resources online. Using your favorite search engine, just search for "high CRI flashlight" or "high CRI LED flashlight", or something along those lines. I was able to find all kinds of things by doing this.
One of the problems in doing night time photography with newer LED flashlights is that many of them, especially cheaper ones, often cast a somewhat bluish or greenish hue, which is exaggerated in the final image during a long exposure. This could also be an issue with iPhone photography as well, but I suspect it would be to a smaller degree. These types of lights probably don't qualify as high CRI lights, though they do come in handy for setting up equipment and other normal flashlight uses. I have a small Nebo LED light that I carry with me almost all the time. I recently used it for doing some long exposure night shooting and the results were pretty bad, and the pictures had a strong blue tone that wasn't able to be color corrected in post. I have another light from Inova that does much better, though Inova lights can get pricey, especially for ones with a high Lumen output.
Speaking of output, the next problem you're likely to encounter is the actual light output of the light itself. I have to admit, Apple has made an amazing LED flash on the iPhone. It is adequate for snapshots out to about 8 feet, and still usable to about 12-15 feet, but not much beyond that. But, for what it is, it's got great light output. It's definitely nothing like having a SpeedLight on a DSLR though. You'll probably want a light that puts out at least 80 Lumens, but 150+ is going to be best. LED lights usually have a narrowly focused beam, so you're probably not going to get a nice "fill" light that you might desire. This is definitely something to consider.
Another thing that I just thought of Since you're planning to use your iPhone are the widely available LED Video light kits for DSLR and other cameras. They're usually a square panel of LEDs designed to fit into a flash shoe on your camera, but they can be easily hand-held by you or an assistant. The video versions usually are ones you can turn on or off and have a good area of fill coverage. Make sure if you go this route to stay away from the LED flash panels, as they'll only flash in coordination with a DSLR. You can also find some cheaper ones that do a great job too. When going this route, pay attention to the Kelvin levels of the lights. I can't really give you a specific Kelvin range to recommend, since that can vary depending on the conditions in which you shoot. But you might want to take a few shots with your iPhone using the on-board flash, and then use Photoshop to determine the color range of the photo, and base your purchase on that. Many LED video light kits include a 3200 Kelvin filter for Tungsten lights (like for shooting in an area with streetlights) or a 4200 Kelvin "soft white" filter, which is probably good for indoors and most other situations. The iPhone will probably be somewhere in the range of 4000 to 6000 Kelvin for soft to warm lighting conditions.
Like I said in my previous post, the lack of ISO and shutter speed control is the one thing that has always bothered me about the iPhone's camera, and I can't say it's specific to any model of iPhone. There are camera apps for the iPhone that supposedly give you more control, but I've found these to be all too limiting. I'd really love to have the ability in future iPhones to be able to take long exposures, at least up to 30 seconds, but longer would be great too! Long exposures isn't going to help freeze the motion of your daughter, since children tend not to sit still for more than a nanosecond, but it's always a desire I've wanted in the iPhone. Maybe it will come in the coming years, but I've learned to work within the limits of the iPhones camera for what it can do. It is what it is. That said, I still don't know of any other mobile phone cameras that come close to the dynamic range of the iPhone's camera, especially the 4S, thought the new Nokia 900 might take the cake in the near future. Only time and testing will tell for sure.
Again, I hope this information and insight helps and gets you pointed in the right direction. Let me know if you have additional questions and I'll be happy to share what knowledge I have!