Does Svellsongur allow Caitlyn to use her activated ability twice?

Ruling: Svellsongur does basically nothing for Caitlyn. While it gives her two copies of her activated ability, you can only activate one of them because activating an ability requires exhausting Caitlyn as a cost, and you cannot exhaust her twice.

Sequence:

• Svellsongur gives Caitlyn two versions of her activated ability
• To activate an ability, you put it on the chain, then pay the cost
• You choose one of the two abilities to put on the chain
• You must pay the cost of exhausting Caitlyn
• Since Caitlyn is now exhausted, you cannot activate the second copy of the ability

Nuances:

• Each copy of the activated ability is its own separate ability
• The ability is "activated" not "triggered" (terminology distinction)