  Fix an odd prime $p$. Let $X$ be a pointed space whose $p$-completed K-theory
$\mathrm{KU}_p^*(X)$ is an exterior algebra on a finite number of odd
generators; examples include odd spheres and many H-spaces. We give a
generators-and-relations description of the
$\mathbf{E}_\infty$-$\mathrm{KU}_p$-algebra spectrum $\mathrm{KU}_p^{X_+}$ of
$\mathrm{KU}_p$-cochains of $X$. To facilitate this construction, we describe a
$\mathrm{K}(1)$-local analogue of the Tor spectral sequence for
$\mathbf{E}_1$-ring spectra. Combined with previous work of Bousfield, this
description of the cochains of $X$ recovers a result of Kjaer that the
$v_1$-periodic homotopy type of $X$ can be modelled by these cochains. This
then implies that the Goodwillie tower of the height 1 Bousfield-Kuhn functor
converges for such $X$.
