Hi thelole,
First, to address the 2 second blackout problem. The high voltage inverter that drives the CCFL monitors the current flowing through the lamps. When the current sensed is incorrect, the inverter shuts down. Two broken lamps implies the sensed current is not what the inverter is expecting, so replacing the transformer will result in some disappointment. Personally, I have seen 10 or more broken CCFL, in Dell Ultrasharp displays with four lamps each. In each case, they appeared to have failed from excessive heat right next to the metal end electrode.
Second, LED strips cannot be directly substituted for CCFL as each requires differing drive requirements. CCFL inverter output can be up around 1.5 kV to 1.8 kV at only a few milliamps. LEDs requires an order of magnitude or so more current at a much lower voltage.
The easiest repair would be to order new CCFL from Mouser or Digikey. If you feel adventurous, it may be possible to substitute lamps from another LCD panel, even if they are a CM or two shorter than the original.
Since you have already disassembled the panel and removed the lamps, you'll have some idea about how fragile CCFL are to handle.