Option for some light between broken fluorescent lights and new LED recessed lights
The option that I use for this type of thing is to add light with portable lighting fixtures. These are extremely handy for adding lots of light in a localized area where you want to perform some work. I use two types.

Halogen Work Light



Picture Source

These output a tremendous amount of light. The wire cage on the front is there to protect from burns as the fixture will get quite hot in use.

LED Worklight



Picture Source

These operate much cooler but also put out less light compared to the halogen type. I selected the LED light with an internal rechargable battery. I can use it plugged in but then can also use it in a portable manner without having to be near an electrical receptacle or dragging an extension cord. The battery operation also comes in extremely handy in cases where it is necessary to work in a dark area because you have had to cut the power at the service panel.