I am trying to send an .png image file from one iOS Device to another over Bluetooth 4.0 LE.
I am able to simple pieces of data like strings, but unable to successfu
BTLE is significantly limited in terms of the amount of data that can be sent both in the characteristics and in any data packet. You should make the connection between the devices and then split the image data up and send multiple small packets. The 2013 WWDC videos will provide you a good overview and code samples.
On iOS 6, BLE allows you to send around 20 byte chunks of data. For an example of how you can slice up the data to be sent and how to use notifications based transmission, download the BTLE Transfer Apple Example application.
To get a better understanding of what speeds you can achieve, I suggest you analyze this diagram http://www.scriptreactor.com/conn_interval-throughput.pdf It shows the achievable speeds as a function of connection interval. iOS does not give you such a fine grained control but this info is still valuable for you to calculate with.