  Performing qubit gate operations as quickly as possible can be important to
minimize the effects of decoherence. For resonant gates, this requires applying
a strong ac drive. However, strong driving can present control challenges by
causing leakage to levels that lie outside the qubit subspace. Strong driving
can also present theoretical challenges because preferred tools such as the
rotating wave approximation can break down, resulting in complex dynamics that
are difficult to control. Here we analyze resonant $X$ rotations of a silicon
quantum double dot hybrid qubit within a dressed-state formalism, obtaining
results beyond the rotating wave approximation. We obtain analytic formulas for
the optimum driving frequency and the Rabi frequency, which both are affected
by strong driving. While the qubit states exhibit fast oscillations due to
counter-rotating terms and leakage, we show that they can be suppressed to the
point that gate fidelities above $99.99\%$ are possible, in the absence of
decoherence. Hence decoherence mechanisms, rather than strong-driving effects,
should represent the limiting factor for resonant-gate fidelities in quantum
dot hybrid qubits.
