What would you do if you were Aragorn after Gandalf had fallen?

[Lord of Angmar]

I doubt I would have followed your course of action. As has been said, taking them to Rohan would take them to close to Saruman and Isengard, and strategically it would make no sense with regards to Frodo and the Ring. There would be no point for Frodo to be in Rohan, as it would be too dangerous for him to assist in the battle of Helm's Deep and also since they were unsure at that point if Rohan remained faithful to Gondor. If I were Aragorn, I would lead them exactly as they went: to Lothlorien, the obvious next choice, and then down the Anduin, making good speed towards both of their possible destinations (Minas Tirith and Mordor) and allowing much-needed thinking time. I don't know if I would have had the power to trust Frodo and Sam going alone to Mordor, but certainly it would be better than having them wait idle somewhere else.