I'm still not convinced that it works.
He had a horse for well over half of his journey. Let's say 800 miles of it. Now, 10 mph on horseback would not seem unreasonable, particularly as most of that stretch was on roads. So, that's 80 hours covering that distance. Say he was travelling for 8 hours a day, which leaves him plenty of time to gather food, sleep etc - that's 10 days.
Then he loses his horse, and has to cover the rest of the journey, say 300 miles, on foot. Let's be conservative and say that he is only able to go at 2mph on foot. It's harder going through this country. That's 150 hours - another 20 days on 8 hours per day.
So, he's got to the land surrounding Rivendell in 30 days. That leaves him searching for the place for 80 days, which is nearly 3 months!!!
That sure is a long time searching.
Edit: To make the maths work.