It's impossible to know for sure!

However there is a few clues based on her close connections.

• Firstly she and her sisters attended Marymount High School, a prestigious private all girls high school based in Bel Air. I underlined private for a solid reason. Private school students are preselected and those with behavioral problems or below average intelligence simply wouldn't get accepted at such a school. Based on this alone, it's safe to say that Kim has at least average intelligence and likely above average intelligence.
• Secondly she is the daughter of two intelligent parents. Robert Kardashian was a highly successful lawyer who was one of OJ Simpson's defence attorneys during his 1995 murder trial. Kris Jenner was the daughter of an engineer and teacher so it's likely that she is quite smart in her own right. Since IQ is largely hereditary, Kim is likely to very intelligent despite her persona.
• Thirdly two of her full siblings did very well academically. Kourtney got accepted at both Southern Methodist University and later transferred to the University of Arizona. To get accepted at SMU you need SAT scores of around 1450 likely placing Kourtney in the top 5–10% academically. Rob Kardashian did even better getting accepted at the USC likely placing him in the top 3% academically.
• Fourthly she is married to Kanye West who is one of the most accomplished rappers of his generation and a terrific entertainer in his own right. West had his IQ tested on at least 2 occasions and most recently tested at 133. Given that most marriages are formed through assortive mating it is unlikely that Kim's IQ is far off 130.

Kim Kardashian doesn't in anyway present herself to be an intellectual or particularly intelligent through her show Keeping up with the Kardashians. However it is far easier for an intelligent person to fake their lack of intelligence than vice versa.

And despite an unpromising start and no obvious talents she succeeded in becoming one of the biggest reality TV stars on the planet. Clearly Kim knows how to recognise and seize opportunities even if it means “selling" her body just for the sake of constant fame and fortune. Personally I think she is marketing genius and world class salesperson.

I mean think about it?

She got millions of people around the world to watch her show and give up their limited time despite it adding no real value to their lives besides a small dosage of mild entertainment.

Based on the above I think it's fair to say that she has an IQ somewhere between 120 and 130.