When I tried to play this prelude some decades ago there was no internet and no way to google and I also had access to music libraries. So I came to same conclusion as Laurence and Tim are advising: I've read it in C, ignoring the 7 sharps.
But what could I read the other day?
Bach recycled some of the preludes and fugues from earlier sources: the 1720 Klavierbüchlein für Wilhelm Friedemann Bach, for instance, contains versions of eleven of the preludes of the first book of the Well-Tempered Clavier.
The C♯ major prelude and fugue in book one was originally in C major – Bach added a key signature of seven sharps and adjusted some accidentals to convert it to the required key.
That's what we are going to do now and writing the harmony (secondary fifths!)
I didn't add the the Roman Numbers of the 4 bars in line 1-4 (it is obvious that it is 1,2,3,4,3,2,1 in the degrees tonic, dominant, supertonic, relative key).
This transcription in C:
In bar 32 he starts with a chain of secondary dominants
B7 -> Em, A7 -> Dm, E7 -> Am, D7 -> G, A7 -> Dm, G7 -> C, D7 - Gm, D7 -> F
we can even extend this secondary chord progressions considering always 2 chords together as "secondary" of the next degree: (iv-V7/) or (ii-V7)
mind that the chords are in 1st (or 3rd) inversion: the 3rd in bass respectively the 7th).
The role and function of secondary dominants is explained here:
What is a secondary dominant chord?
Now it's quite easier to read and understand the progression in C# major:
And if we look at the transcription of Franz Kroll in Db we will understand even better the functions of the secondary dominants. That's why I believe it is easier to read it in Db major: look at the 1st chord of the V7/X (secondary dominants)! We read C7 -> F ... etc. this is much easier to read and analyse than B double sharp -> E#