I'm afraid you are mixing up two categorically different issues and inferring what is not right.
First, there was a case in California , perhaps, very much politically motivated. Please acquaint yourself about the circumstance of the case, perhaps, to reach to a different conclusion. Nawaz was fully behind the case and so was, later, claimed by Miss White herself. However, that is not the real issue.
What is important is the fact that IK denied her claim in the court, through his attorney, asserting that that was impossible. That is a typical case of 'he said, she said'. The judgment resulted only because of a certain law in California, not because of any positive proof; there wasn't any biological test.
The timing of the case fell in the middle of the election campaign and, Nawaz machine was funding and driving the case; and they dictated its timings; they published juicy reports in local media consisting of the damaging claims by the attorney, who was hired by Nawaz, representing Miss White along with pictures, the one I recall had the attorney standing tall in front of the court with a beautiful looking lady close on his side – truly a front page material!; their target was to malign a budding challenge coming from inside Punjab; and no qualm in accepting the fact that Nawaz machine's exercise succeeded in its nefarious aims, causing a permanent dent to the viability to PTI for a quite long period.
It was ill-advised for IK to go to California in the middle of the events and testify in person, I suppose because of two reasons;
a. his travel to California for this purpose implied that the Pak press would prominently highlight his whole visit/court proceedings with an obvious tilt and, thus he would had to bear further political damage;
b. the judgment in California's lower court would not had been binding for him and couldn't had resulted in any financial and legal consequences for him. As per the legal statue of the lower court, if he were to fail to testify in person, the judgment would be automatically awarded favoring the complainant without any further proceedings. And that was the result.
Thus, the nature of the case is still 'he said, she said' while, it is on record that she accepted that she was used/abused for political purposes. Further, at that time she was emotionally and financially in shambles, suspected of heavy drug use, which, perhaps, sooner, afterwards, led to her death. Further, IK's attorney put forth a number of arguments, based on the timings and, if I remember, including a testimony of her friend, showing the impossibility/implausibility of such occurrence and mentioned another man as her probable father. Still, for us, these are all peripheral issues - the matter remains that - she made a claim and he denied that strongly; that's end of it.
Second is the issue of the girl child;
Her mother died when she was very young. I do not know her; however it is about certain to assume that, lacking any family support, the unfortunate kid would find emotional and psychological comfort and insurance by adhering to her mother's claims. I assume, she might had insisted to the claim for to assure herself and find her identity - that's very natural. In such circumstances, considering the welfare of the lonely kid, it would be much humane and advisable to adopt or, at least, provide some emotional space to the child. Adoption is a common phenomenon, if one can afford – and the best option, if IK were some truly western citizen. Of course, IK being in politics, cannot afford this kindness and benevolent attitude, however, his career’s restrictions cannot possibaly convince his sons or ex-wife to shun a girl who, I assume, claims to be their relative, and not to befriend with her or not to be close to her - why should they had done so! If they did that, that' s very good. But, to infer from such specific circumstances that the IK claims were wrong or he had lied in the court- I’m afraid, that's logically not right. I'll only say that if his sons and Jemima were open giving her a space and comfort, that's very kind of them and a similar behavior is expected from the majority of people in the west;--- that would make her their real step-sister-no way; such relationship, if there is the one, doesn’t furnish any kind of evidence.
No doubt, flooded by daily dose of abuse of Ik by the political opponents, one would be inclined to find ill-will for IK than to see some good in his family. That's all politics!
If one were to believe into the incessant propaganda of Noonies, how come a top level fast bowler could be so irresponsible to indulge into drugs and alcohol, only in the world of Talaals! Once Botham claimed that he never had a sip of alcohol throughout his professional career, let alone the drugs. Yes the top athletes indulge them into all sorts of diets but to enhance their stamina but not to dump them. Only in the world of Noonies an abuser of cocaine and churs, climbs running up the mountains at this later age.