How to retrieve a random word of a given length from a Trie

I have a simple Trie that I'm using to store about 80k words of length 2 - 15. It works great for checking to see if a string is a word; However, now I need a way of getting a random word of a given length. In other words, I need "getRandomWord(5)" to return a 5 letter word, with all 5 letter words having an equal chance of being returned.

The only way I can think of is to pick a random number and traverse the tree breadth-first until I've passed that many words of the desired length. Is there a better way to do this?

Possibly unnecessary, but here's the code for my trie.

class TrieNode {
    private TrieNode[] c;
    private Boolean end = false;

    public TrieNode() {
        c = new TrieNode[26]; 
    }

    protected void insert(String word) {
        int n = word.charAt(0) - 'A';
        if (c[n] == null)
            c[n] = new TrieNode();
        if (word.length() > 1) {
            c[n].insert(word.substring(1));
        } else {
            c[n].end = true;
        }
    }

    public Boolean isThisAWord(String word) {
        if (word.length() == 0)
            return false;
        int n = word.charAt(0) - 'A';
        if (c[n] != null && word.length() > 1)
            return c[n].isThisAWord(word.substring(1));
        else if (c[n] != null && c[n].end && word.length() == 1)
            return true;
        else
            return false;
    }
}

Edit: The marked answer worked well; I'll add my implementation here for posterity, in case it helps anyone with a similar problem.

First, I made a helper class to hold metadata about the TrieNodes I'm using in the search:

class TrieBranch {
    TrieNode node;
    int letter;
    int depth;
    public TrieBranch(TrieNode n, int l, int d) {
        letter = l; node = n; depth = d;
    }
}

This is the class that holds the Trie and implements the search for the random word. I'm kind of a