Consider the following simplified example of the same phenomenon, which many students find clarifying.

Let $f(n)=1$, if there are $n$ consecutive $7$s in the decimal expansion of $\pi$, and otherwise $f(n)=0$. Is this function computable? 

A naive attempt to compute $f(n)$ would simply proceed to search $\pi$ for $n$ consecutive $7$s. If found, the algorithm outputs $1$, but otherwise....and then the naive algorithm doesn't seem to know when to output $0$, and so students sometimes expect that $f$ is not computable.

But actually, $f$ is a computable function. If it happens that there are arbitrarily long sequences of $7$s in the decimal expansion of $\pi$, an open question, then $f$ is the constant $1$ function, which is certainly computable. Otherwise, there is some longest sequence of $7$s in $\pi$, having length $N$, and so $f$ is the function that is $1$ up to $N$ and then $0$ above $N$. And this function also is computable, for any particular $N$.

So the situation is that we have proved that $f$ is computable by exhibiting several algorithms, and proving that $f$ is definitely computed by one of them, but we don't know which one. (In fact, $f$ is linear time computable.) So we have proved that $f$ is a computable function, but by a pure existence proof that merely shows there is an algorithm computing $f$, without explicitly exhibiting it. 

It seems to be the same phenomenon in your case, where you have a computable function, but you don't know which algorithm computes it.