Reading through the problem, my initial thought was that Alice was right. That many strings of 5(and more)or statistically quite unlikely and, in any case, the letters are very unlikely to be "randomly generated text". I did some more investigations to check- I found the differences between the ends of the identical strings of letters.
My Results
Green 35
Red 115
Yellow 795
Dark green 480
Grey 670
Pink 550
Turquoise 490
The noticable thing about these results is that they are all divisible by 5.
This adds strength to Alice's argument, making it less likely to be coincidental.
I decided to try putting the letters into columns of 5. Though the frequency analysis shows that it is not a letter for letter code, it could be a polyalphabetic code (where more than one ciphers are used in alternation).
I found the frequencies for the first column, made them into a graph and matched it with the English frequencies. It was quite clear from the patterns that it was a ciesor shift, with plaintext a matching with cipher text n. Feeling encouraged, I substituted that column and found the frequencies for the others. After quite a long time trying to get my computer to do what I wanted it to, I arrived at an answer which (mostly) made sense. Filling in the word gaps gave the solution:
I AM ALREADY FAR NORTH OF LONDON AND AS I WALK IN THE STREETS OF PETERSBURGH I FEEL A COLD NOR THE RN BREEZE PLAY UPON MY CHEEKS WHICH BRACES MY NERVES AND FILLS ME WITH DELIGHT DO YOU UNDERST AND THIS FEELING AT HIS BREEZE WHICH HAS TRAVELLED FROM THE REGIONS TOWARDS WHICH I AM ADVANCING GIVES ME A FORETASTE OF THOSE ICY CLIMES INSPIRITED BY THIS WIND OF PROMISE MY DAYDREAMS BECOME MORE FERVENT AND VIVID I TRY IN VAIN TO BE PERSUADED THAT THE POLE IS THE SEAT OF FROST AND DESOLATION IT EVER PRESENTS ITSELF TO MY IMAGINATION AS THE REGION OF BEAUTY AND DELIGHT THERE MARGARET THE SUN IS FOREVER VISIBLE ITS BROAD DISK JUST SKIRTING THE HORIZON AND DIFFUSING A PERPETUAL SPLENDOUR THE REFOR WITH YOUR LEAVE MY SISTER I WILL PUT SOME TRUST IN PRECEDING NAVIGATORS THE RE SNOW AND FROST ARE BANISHED AND SAILING OVER A CALM SEA WE MAY BE WAFTED TO A LAND SURPASSING IN WONDERS AND IN BEAUTY EVERY REGION HITHER TO DISCOVERED ON THE HABITABLEGLOBEITSPRODUCTIONS AND FEATURES MAYBE WITHOUT EXAMPLE AS THE PHENOMENA OF THE HEAVENLY BODIES UNDOUBTEDLY ARE IN THOSE UNDISCOVERED SOLITUDES WHAT MAY NOT BE EXPECTED IN A COUNTRY OF ETERNAL LIGHT AI MAY THERE DISCOVER THE WONDROUS POWER WHICH ATTRACTS THE NEEDLE AND MAY REGULATE ATHOUS AND CELESTIAL OBSERVATIONS THAT REQUIRE ONLY THIS VOYAGE TORENDER THEIR SEEMING ECCENTRICITIES CONSISTENT FOREVER I SHALL SATIATE MY ARDENT CURIOSITY WITH THE SIGHT OF A PART OF THE WORLD NEVER BEFORE VISITED AND MAY TREAD A LAND NEVER BEFORE IMPRINTED BY THE FOOT OF MAN THE SEARE MY ENTICEMENTS AND THEY ARE SUFFICIENT TO CONQUER ALL FEAR OF DANGER OR DEATH AND TO INDUCE ME TO COMMENCE THIS LABORIOUS VOYAGE WITH THE JOY A CHILD FEELS WHEN HE EMBARKS IN A LITTLE BOAT WITH HIS HOLIDAY MATES ON AN EXPEDITION OF DISCOVERY UP HIS NATIVE RIVER BUT SUPPOSING ALL THESE CONJECTURES TO BE FALSE YOU CANNOT CONTEST THE INESTIMABLE BENEFIT WHICH I SHALL CONFER ON ALL MANKIND TO THE LAST GENERATION BY DISCOVERING A PASSAGE NEAR THE POLE TO THOSE COUNTRIES TO REACH WHICH AT PRESENT SO MANY MONTHS ARE REQUISITE OR BY ASCERTAINING THE SECRET OF THE MAGNET WHICH IF AT ALL POSSIBLE CAN ONLY BE EFFECTED BY AN UNDERTAKING SUCH AS MINE
It might not be perfect, but it gets across the message. This shows that the repeated messages were parts of this message enciphered under the same part of the key, which is a lot more likely than random letters.