I think you'll find it difficult to find an apartment for such a short time, unless you find someone who is willing to sublet their apartment for a few weeks. Apartments are in high demand in Delhi, so landlords tend to be reluctant to engage in short term contracts.
It might be a good idea to try and find an hotel or apart-hotel for the period that your wife will be visiting you. You can read more about the situation on the Expat Arrival's Accommodation in Delhi page.
You'll often find good last minute deals online.In fact, due to high demand for apartments you'll find that landlords will charge more to let out their properties on a short-term basis. Therefore it might even be cheaper for you and your wife to stay in a hotel.