After I attach Svellsongur to Jayce Brilliant Inventor, if I play the first gear for that turn, can I ready two things?

Yes — you can ready two things.

Svellsongur, when attached to Jayce, Brilliant Inventor, copies his printed text back onto Jayce, giving him two instances of "the first time you play a non-token gear each turn, you may ready something besides me that's exhausted" (the same duplication effect confirmed in FAQ #376/#7915 for Azir and Aphelios). When you then play your first non-token gear that turn, each instance independently sees its trigger condition met and goes on the Chain, so you get two separate "ready something besides me" effects — one per instance. Each instance is limited to one ready, so you can ready up to two different exhausted things (still none of them Jayce himself).