This

clear;

Is not a defined Mathematica function. That is indicated because it is shown in pale blue in your screen capture.

Perhaps you meant

Clear["Global`*"];

and wish it to erase all prior assignments. You can look that up in the documentation.

But I do not know that adding "clear;" to your code caused any problems. Probably it did nothing.

Then you define

ratio2[N2_, k1_] := (N2 K1)/K1prime

and it is not obvious to me if the calculation of ratio2 makes any use of the second argument or simply ignores it.

You do have multiple uses of k1 and K1 mixed together prior to this. I attempted to use Trace[] to track down exactly how the numeric values from FindRoot are used within the search process in your code and could not come up with a sufficiently simple explanation that I thought it would help.

Each of us makes changes and shows the other what we get and the other does the same and this repeats. But I am not sure we are making progress with this, it works for me without errors and it doesn't work for you with errors, again and again.

Attachments:

Like that?

ratio2[N2_, K1_] := (N2 K1)/K1prime

Because it doesn't work either for me

There seem to be a handful of tiny typos in the code. Guessing how to fix those gives

In[1]:= Subscript[R, p] = 0.1;
Subscript[R, s] = 55;
Subscript[\[Omega], p] = 1;
Subscript[\[Omega], s] = 1.1;
Subscript[\[Omega], c] = Subscript[\[Omega], p];
m = k^2;
\[Epsilon] = Sqrt[10^(0.1 Subscript[R, p]) - 1];
k = 1/Subscript[\[Omega], s];
kprime = Sqrt[1 - k^2];
K = EllipticK[m];
Kprime = EllipticK[kprime^2];
k = Sqrt[(10^(0.1 Subscript[R, p]) - 1)/(10^(0.1 Subscript[R, s]) - 1)];
k1prime = Sqrt[1 - k1^2];
K1 = EllipticK[k1^2];
K1prime = EllipticK[k1prime^2];
N1 = (K1prime K)/(K1 Kprime);
N2 = Ceiling[N1];
ratio1 = K/Kprime;
ratio2[N2_, k1_] := (N2 K1)/K1prime;
k1nouveau = FindRoot[ratio2[N2, k1] == ratio1, {k1, 0.00027140}, MaxIterations -> 200]

Out[20]= {k1 -> 0.000177117}

Please check all of this very carefully in a freshly started notebook