There are some factors that other answers have not covered.
The main claim centers around Illinois. Even though flipping Illinois would not have changed the overall result, there are some things to consider.
First of all, let's concede that Chicago politics is a filthy/dirty as it gets (Report:Chicago most corrupt city in the U.S), and was/is largely controlled by the Democratic political machine, and that Mayor Daly controlled things as much as any politician in recent memory. Does that mean that it's a given that Illinois was stolen? No.
His father, the late Mayor Richard J. Daley, built the once-mighty machine that doled out jobs and favors in exchange for support for Democrats on Election Day. He was never charged with criminal wrongdoing, but several of his high-ranking aides were sent to prison for political patronage.
NBC News, Crime and Courts:Illinois has a long legacy of public corruption
Kennedy won the Chicago vote by a huge margin, but, demographically, that margin was not unprecedented, and Kennedy had certain demographic appeal that was unprecedented -
Kennedy’s margin in Cook County is sometimes overstated as 450,000 votes (or more precisely 456,312), but that figure is limited to the city of Chicago. Cook County also includes a large suburban area, which Nixon won. Across all of Cook County, Kennedy’s margin was 318,736 votes.
For comparison, Dwight Eisenhower had won Cook County by a similar margin in 1956 (315,402 votes). In 1964, Lyndon Johnson would win Cook County by more than twice as many votes (641,463).
Of course, 1956 and 1964 were landslides. How did Kennedy rack up such a margin in a squeaker like 1960? The key is Cook County’s demographics, which were very favorable to Kennedy. Thirty-nine percent of Cook County residents were Catholic, and 20 percent were black. Black and Catholic voters heavily favored Kennedy, who was Catholic and had recently helped to secure Martin Luther King Jr.’s release from a Georgia prison.
There were two Cook County recounts, because the overall results were so close. One was immediate, so the results could be certified, and another was after Kennedy was sworn in. Both narrowed the margin, but neither flipped it. Now, that doesn't mean that there were no shennanigans, but one of the arguments pointing to fraud is the claimed lack of recount.
One way to bring fraud to light is with a recount, and there is a persistent myth that there was no recount in Cook County. In fact, there were two recounts — one in November 1960, before the state vote was certified, and one after Kennedy’s inauguration in 1961.
According to Kallina’s history, both recounts whittled Kennedy’s margin in Cook County, but not by enough to erase his lead statewide.
Dissatisfied Republicans pointed out that some forms of fraud — such as buying or coercing votes — were not the kind that could be corrected by simply recounting the votes. Democrats countered that the downstate counties, which hadn’t been recounted, might have had fraud favoring Nixon, but Democrats did not offer evidence to support their accusation.
WaPost: Here's a voter fraud myth - Richard Daly stole Illinois for Kennedy
Why didn't Nixon demand a recount to rectify the fraud? One item that is accepted by historians, along with the probable pro-Kennedy fraud in Chicago, is that there was also rampant pro-Nixon fraud in downstate Illinois. While a recount might bring the Daly abuses to light, it would also expose similar fraud going on for Nixon, and ultimately would probably not be worth that examination for a result that would not change the outcome of the election.
It is true that Vice President Nixon did not demand a recount in Illinois, despite allegations that Democratic voter fraud in Cook County was rampant, possibly even greater than Republican fraud downstate. Mr. Nixon patriotically stated (after some delay) that his decision recognized the need for national unity and stability during the Cold War. But, to put it in perspective, the following should be noted:
(1) Mr. Nixon decided to refrain from challenging the result only after, as stated in his memoirs, he had "looked into the legal aspects of the situation" and found no basis on which he could obtain a timely victory.
(2) A successful lawsuit switching Illinois from Kennedy to Nixon would not have ended Kennedy's electoral vote majority, even if Alabama's six "faithful electors" had joined their segregationist colleagues in voting for Harry Byrd. Kennedy's Texas margin was too large to reverse, even had a recount been available.
(3) Having trailed in the popular vote on Election Day (depending on how Alabama's vote was measured), Mr. Nixon was ill-positioned to claim, even if some ballots should have been ruled out, that the will of the American people had been thwarted.
WaPost: Not quite 1960