How can I build a fixed point theorem argument in pure strategies?

To prove that that some given combination of strategies is a Nash equilibrium you don't need to use a fixed-point theorem (such as Brouwer's or the fixed-point theorem for contractions on Banach spaces). What you do have to do, is check that they are best responses to one another. This true for mixed and pure strategies.

You also seem to be asking how you can use the Banach fixed-point theorem to prove existence of a pure strategy equilibrium. As @Giskard pointed out, this is doomed to failure without restrictions on the game, as not every game admits a pure strategy equilibrium.

If you wanted to try anyway: Let $S_{i}$ denote $i$'s set of pure strategies. Let $X = times_{i} S_{i}$ and define $T colon X to X$ as follows: $T(x) = (T_{i}(x))_{i}$ where $T_{i}(x)$ is some pure strategy of the best response correspondence of agent $i$ to the strategy profile $x_{-i}$ of the others. (If $T$ is supposed to be a contraction on $X$, then it has to map to $X$, not the power set. Note, however, that this selection is irrelevant if the game happens to be such that best responses are always unique.) A profile of pure strategies is a Nash equilibrium if it is a fixed-point of $T$. Conversely, for every Nash equilibrium $x$, there is a way to define the selection from the correspondence such that $x$ is a fixed-point of the thusly defined mapping $T$.

The difficult in this approach thus lies in finding a selection from the best response correspondence together with a metric such that $X$ is a complete metric space on which $T$ is a contraction.

But this is not possible in general: Not every game admits a pure strategy Nash equilibirum, as pointed out by @Giskard. Not only that: Not every game which does admit such an equilibrium has a unique one (consider the standard coordination game, where best responses are always unique but which has multiple pure strategy equilibria). There is simply no good reason (at least one that I can see) why, without restrictions on the game, best responses would allow you to define a contraction. (But perhaps someone knows a class of games where this is actually the case.)

All of this is to say that there is no existence result for pure strategy Nash equilibria in the generality that you seem to describe. Results for games in which such pure equilibria can be proven to existence abstractly (e.g. via Kakutani's fixed-point theorem) rely on more special features of the game (e.g. in Athey (2001), certain single-crossing properties are key). What tools might be useful to you will depend, to a degree, on the details of your problem.