How did Frodo get a second sword?

[The Squatter of Amon Rûdh]

'Look!' he cried; and stooping he lifted from the ground a black cloak that had lain there hidden by the darkness. A foot above the lower hem there was a slash. 'This was the stroke of Frodo's sword,' he said. 'The only hurt that it did his enemy, I fear; for it is unharmed, but all blades perish that pierce that dreadful King.

From Flight to the Ford

Obviously Frodo only managed to slash the cloak, which was then abandoned in the confusion. Since Tolkien only describes Frodo striking at the feet of his enemy, and since Strider specifically tells everyone that he missed his mark, I see no discrepancy here at all.