What should one do if he swore to perform the Tahajjud prayer but was unable to sleep, and consequently only performed the Qiyam al-Layl prayer?

"Tahajjud" refers to praying at night after waking up from sleep, whereas "Qiyam (definition) al-Layl" refers to praying at night in general. If the intent of "Tahajjud" was merely "Qiyam al-Layl" before dawn, then there is no breaking of the oath for someone who performed Qiyam al-Layl. However, if the intent was specifically the Tahajjud prayer in the last part of the night, then if the time of the prayer was the intended time, there is no breaking of the oath. But if it was before that, the one who swore the oath must offer expiation for breaking an oath. A Muslim should not swear an oath on something for which there is ample flexibility.