Finding the most 'unique' word
APL (56)

{⎕ML←3⋄⊃{⍵,⍴∪⍵}¨W[⍙]⍴⍨↑+/∆∘.=∆←∆[⍙←⍒∆←↑∘⍴∘∪¨W←⍵⊂⍨⍵≠' ']}


This is a function (question says that's allowed) that takes a string and returns a matrix of words and unique lengths.

Usage:

      {⎕ML←3⋄⊃{⍵,⍴∪⍵}¨W[⍙]⍴⍨↑+/∆∘.=∆←∆[⍙←⍒∆←↑∘⍴∘∪¨W←⍵⊂⍨⍵≠' ']}'The quick brown fox jumps over the lazy dog.'
quick 5
brown 5
jumps 5


Explanation:


⎕ML←3: set migration level to 3 (so that ⊂ is partition instead of enclose)
W←⍵⊂⍨⍵≠' ': store in W the given string, where each partition consists of non-whitespace characters.
⍙←⍒∆←↑∘⍴∘∪¨W: get the amount (⍴) of unique (∪) elements in each part (¨) of W, and store these in ∆, then get the sort order when sorted downwards on this (⍒) and store that in ⍙.
∆[⍙...]: sort ∆ by ⍙, so now we have the unique lengths in order.
∆∘.=∆←∆: store the sorted ∆ back in ∆, and see which elements of ∆ are equal.
↑+/: sum the rows (now we know how many elements are equal to each element) and then take the first item (now we know how many elements are equal to the first element, i.e. how many of the words are tied for first place.)
W[⍙]⍴⍨: sort W by ⍙, and take the first N, where N is the number we just calculated.
{⍵,⍴∪⍵}¨: for each of these, get the word itself and the amount of unique characters in the word
⊃: format as matrix