No, the interaction of these two rules does not allow you to draw and attack with two daggers using TWF.
(See @Medix2’s answer for how you might achieve the desired result anyway, though.)
The Two-Weapon Fighting rules (as you’ve quoted) require you to already have both weapons drawn (emphasis mine):
When you take the Attack action and attack with a light melee weapon that you’re holding in one hand, (…)
The “holding in one hand” is a qualifier for both the first and the second weapon, meaning that at the start of this attack combo, you need to have both weapons in one hand each.
The Thrown Weapon Fighting fighting style allows you to draw them as part of the Attack action, which means that when you make attack #1, you do not have weapon #2 in your hand.
This concludes in grammatical nitpickery, because Two-Weapon Fighting specifies that the conditions (both weapons in a hand each) have to be met when making the first attack. Since this (implicitly) excludes drawing them within the action, you cannot use Two-Weapon Fighting in this case.
However, as a DM, I personally would absolutely let this slide. You are already taking a niche and likely non-optimal route, so why not?