This chapter might have been difficult for you, but I believe you mostly pulled it off. My few quibbles are minor:
The change from Eowyn's point of view to Lothiriel's doesn't flow; when it happens, you may want to widen the space between paragraphs or use asterisks to mark the change.
Secondly, this line here caught my attention as feeling rather too modern for the tone you've set and the general style of your writing:
"You may be right. I was furious earlier because poor Ioreth tried to tell me things are returning to normal, because we received a new supply of food that included eggs," Lothíriel admitted grudgingly. "But I want to stop thinking about all that.[...]"
Last, the ending feels rather abrupt (from when Lothiriel goes to ask Eowyn to journey with her to Ithilien and on), so you might want to fill it out a little more...
Apart from that, I very much enjoyed this chapter - the dream sequences were especially nice, as was the moment where you described Faramir & Eowyn's hair mingling in the wind. You took a classic moment and made it fresh - kudos!