  We study the $p$-adic loop group Iwahori-Hecke algebra $\mathcal{H}(G^+,I)$
constructed by Braverman, Kazhdan, and Patnaik and give positive answers to two
of their conjectures. First, we algebraically develop the "double coset basis"
of $\mathcal{H}(G^+,I)$ given by indicator functions of double cosets. We prove
a generalization of the Iwahori-Matsumoto formula, and as a consequence, we
prove that the structure coefficients of the double coset basis are polynomials
in the order of the residue field. The basis is naturally indexed by a
semi-group $\mathcal{W}_\mathcal{T}$ on which Braverman, Kazhdan, and Patnaik
define a preorder. Their preorder is a natural generalization of the Bruhat
order on affine Weyl groups, and they conjecture that the preorder is a partial
order. We define another order on $\mathcal{W}_\mathcal{T}$ which is graded by
a length function and is manifestly a partial order. We prove the two
definitions coincide, which implies a positive answer to their conjecture.
Interestingly, the length function seems to naturally take values in
$\mathbb{Z} \oplus \mathbb{Z} \varepsilon$ where $\varepsilon$ is
"infinitesimally" small.
