Before we consider the effects of a surviving Mughal empire, we first must figure out how the Mughals are going to keep their empire together for a few extra centuries. For starters, they needed to establish proper and commonly accepted rites of succession to the throne, rather then just letting all the claimants battle it out in a repetition of civil wars every time an emperor dies (and in some cases, the battles started well before the reigning emperor was finally dead). The Mughals were actually quite fortunate that between Babur's rise in the 1520's and Aurangzeb's death in 1707 they had the benefit of talented and ruthless leaders who were able to secure the throne relatively swiftly. After Aurangzeb, the succession just became a Darwinian free-for-all, and Mughal central authority suffered immensely as a result (the fact that few of the Mughals after Aurangzeb possessed the qualities necessary for leadership to degrees comparable to their predecessors didn't help matters).
More importantly, though, the Mughals are going to have to come to terms with one simple, unavoidable fact: that India is above all else a Hindu country, and that traditional Muslim attitudes toward "idolatry" simply will not fly when over 80% of your population (at least) are fiercely devoted to their 330 million gods. Aside from Ackbar, few of the Mughals were willing to consider giving the Hindus any more tolerance then the "privilege" of being allowed to pay the jizya (rather then simply being enslaved or massacred, the traditional Muslim proscription for idolaters). Some form of religious plurality with Hinduism, which was nigh-unthinkable to the more conservative Muslims like Aurangzeb, wound have gone a long way towards placating Hindu warrior groups like the Marathas and the Rajputs. That also would have helped keep the general population loyal to the throne against the threat of Europeans, who, (at least at first,) seemed more tolerant of Hindusim then the Muslims.