If you pictures stop saving to your device you have two options. You can delete some pictures to create more memory space on your device, or you can insert a memory card. Using a memory card allows you to save a large amount of photos to your device. Make sure that you purchase the correct memory card for your device.
Your devices battery may not be charged. If the device still won't power on after the battery has been connected to the battery dock and fully charged, you may need to replace the battery. If you are still have problems refer to the Troubleshooting guide: Sony Cyber-Shot DSC-TX55 Troubleshooting.
If your flash on your device is not working, you will want remove the battery and memory card and reinsert both after a few seconds. You will then want to turn on your camera and place it on Auto Mode. Try seeing if the flash works now by pressing the shutter button on to take a picture.