If I kill my Brynhildr with Headless Resurrection, can I then play her again?

No, you cannot use Heedless Resurrection to kill your Brynhir Thundersong and immediately play that same Brynhir from your trash.

According to the rules of game timing and targeting (specifically Rule 355.15), this interaction fails because you must choose your target before paying the costs of the spell.

The Sequence

1. Play the Spell: You announce that you are playing Heedless Resurrection.
2. Make Choices: You must choose a target unit from your trash to be resurrected. At this moment, your Brynhir Thundersong is still on the board and is not in your trash, so it is not a valid target.
3. Lock Targets: Targets are locked in during Step 2. You cannot change your selection once the spell has begun the process of being played.
4. Pay Costs: You then pay the additional cost of the spell by killing your friendly Brynhir Thundersong, moving it to the trash.
5. Resolution: Because your target was already locked in during Step 2, you cannot retroactively choose the Brynhir that just arrived in the trash.

Sources: RiftJudge FAQs #9523 and #10168.