  We consider fourth order singularly perturbed boundary value problems with
two small parameters, and the approximation of their solution by the $hp$
version of the Finite Element Method on the {\emph{Spectral Boundary Layer}}
mesh from \cite{MXO}. We use a mixed formulation requiring only $C^{0}$ basis
functions in two-dimensional smooth domains. Under the assumption of analytic
data, we show that the method converges uniformly, with respect to both
singular perturbation parameters, at an exponential rate when the error is
measured in the energy norm. Our theoretical findings are illustrated through
numerical examples, including results using a stronger (balanced) norm.
