If I was presented with this dissected antenna and asked to estimate the frequency, I would do the following:
(1) Look at the coax feed line and if it is RG-58 or better then I would use VF=0.77 and if it has a thinner core then I would use VF=0.65.
(2) Measure the length of the sleeve balun. Guess that it is probably 1/4 wave.
(3) Straighten out the helical coil and measure its length. Usually 1/4 wave or 1/2 wave.
(4) I would then try and find a number that works for (2) and (3) as the wavelength in the coax.
(5) I would then increase it using the VF to estimate wavelength in vacuum.
(6) I would then calculate the frequency and then look at known bands.
For the coil I looked at the OD (about 5mm). My guess is that it was wound onto a 4mm or 4.5mm form. Because of the straight part soldered to the coax core and the diameter range (4mm to 5mm) I estimated the effective length on the notes at 12cm to 13cm. I would use (3) above to be more precise.
I think that the overall length of your “dipole part connected to the center wire” is the key to the resonant frequency.
You are correct that a “1/2 wave center-fed” dipole is normally considered to have a 1/4 wave length on each side. I posted the picture because I was interested to see that the upper coil is 1/2 wavelength.
It’s almost 40 years since I studied electrical engineering and telecom at university and since then I have cheated on VFs (like everyone else) by either looking at the cable manufacturer’s datasheets or by assuming 0.85 (LMR400), 0.77 (RG58) or 0.65 (cheap/thin/UTP/etc). There’s a small article on wikipedia about VFs and, as usual, google has lots of links.
Actually I cheat even worse… after a working life in telecom, I just buy the cables and antennae pre-made and matching from good manufacturers. I read this topic because I really admire all the brave DiY folks who build their own antennae and then try to understand the results.