I've had good luck with this. The 50/50 Hitlights (or similar) waterproof LED strips and a transformer are fairly easy to install anywhere. You can cut the strips (seal the ends where cut & soldered with clear waterproof tape or plastic dip.) For a non-waterproof solution it's easier, you can simply cut and use the clip-on wire ends.
You'll want to remove the ballast and wiring from inside the fluorescent tube fixture, and install an outlet instead, so that a plug-in transformer can be used.
Cut the strips to the 4 foot lengths, and use four strips per 2 tubes. This will give the approximate equivalent light. The small wires you have removed which go from the ballast to the tube ends of the old fixture can be used to go from the transformer to the strip ends. Be sure to connect the positive and negative leads properly. You won't need much wire. as your transformer will have a 12 volt pigtail wire (where you would plug in any 12 volt accessory.) You can simply cut off the end of the 12 Volt plug-in end (end of the wire coming from the transformer) and use (waterproof in this case) wire nuts to cover where you are splicing 4 positive and 4 negative wires together for the 4 strips. Strip insulation back about 3/4 inch and twist together. You could also use a shrink-wrap type insulator over the bare wire ends. I do recommend soldering the small wire ends onto either the strips, or the connectors, and placing a dot of solder over the wires before putting them in the wire nuts as well. A very tiny amount of solder will be used, and a steady hand will be needed. Practice this on some spare material if you haven't soldered before.
You can also purchase splitters that simply plug together, but I've found it's just as easy to use the wires themselves.
Since most online retailers are selling 5 meter (just over 16 foot) rolls of the 50/50 strips, for about $10 (US) the major cost is the transformer. You'll be able to get 4 strips plus a spare section for each roll. Check the rating on the transformer, some of them will run much more than one 16 foot roll of LED lights, possibly 3 or 4 or more, so you won't need a transformer for each 16 feet of LED strip.
What I do is use the metal (behind where the fluorescent tubes were, on the outside of the housing) to simply put as many strips as you prefer, 2 per where there was one tube, or 4 if you want something twice as bright) and simply use the adhesive backing from the LED strips to mount them in rows.
For the house current, the ground wire (in 120V US) from the wall isn't needed, as LED strips do not need to be grounded. However, you will wish to wire the outlet as a standard grounded 3 prong outlet to code. Often the transformer does have a ground or this will allow you to use either grounded or non-grounded plugs (for transformers or anything else) in the same area if needed, say an extension cord to a power tool. If this is a fixture switched at the wall, you won't have any trouble, as you're wiring this as a switched outlet the same as a plug-in lamp.
If you're not confident you can do this hire an electrician.