A LED is not monochrome. It has band width of around 50nm. You need a light source with a band width of an angstroms or less ( 1nm = 10 angstroms) The CFL bulbs works because of the gases inside the bulb produce a number of monochrome emission of different wavelengths but they are discreet and very narrow. The white coating in bulb partially diffuses them to make the light look white but not enough that the light will still produce interference fringes of different colors. That is one reason why the fringes your see are low in contrast because there are a numbers of them of different colors right next to each other. The other issue is that if the diagonal your testing is aluminized. There is large difference in reflectivity between that surface and the bare glass surface of the reference flat.
So what can you do to increase the contrast ? First you want test with a monochrome source. The simplest way to do that is to filter the light from the CFL bulb so your only viewing one wavelength. That is simple to do. Just look thru a green, red or yellow filter when your doing the test. The gas emission in the CFL bulb are far enough way from each other in wavelength that a simple filter, of the color you want to use will isolate that wavelength from the others. I have used clear red or green candy wrappers to look thru to increase contrast.
The second is harder to do. That is to get a better match in reflectivity between the aluminized surface and the bare glass. That requires the surface on the reference flat be partially coated to increase it's reflectivity. I have flats that are coated for this.
The good news is that all you need to do is see the fringes, even faintly to judge the flatness. What is critical thou is the surfaces are very clean and that when you see the fringes, you want no more then 5 of them being visible. This greatly increases the sensitivity of the test. When more are visible it is hard to judge how much of any they depart from straight and when many are visible a surface that is waves from flat will show straight fringes.