The Kansas City Royals' real prize for coming from two games down to win the World Series didn't come until yesterday. The team went to the White House to meet President Reagan in the Rose Garden, then moved on to Capitol Hill to hobnob with seemingly every politician from Missouri and Kansas who ever heard of baseball.

Reagan recalled a 1952 movie, "The Winning Team," in which he played a pitcher, Grover Cleveland Alexander, for the St. Louis Cardinals, the team Kansas City rallied to defeat in the recently concluded seven-game Series. "I pitched in a World Series, but I was with the Cardinals," Reagan said. "(But) I had an edge on all of you in the sense that I knew the script in advance, so I knew it was going to come out right."

Reagan went on to tell the Royals, "You've proved to America what a never-say-die spirit can do. Even after losing the first two games of the Series at home, you met the challenge and you kept America in suspense for seven full games and rallied to bring the World Series trophy to Kansas City.

"I always like to point to experience as a major element of success. But it's hard in this case when you add up the ages of three of the five starting pitchers and come up with fewer birthdays than I've seen."

When Reagan asked, "What do you do for an encore?" Royals Manager Dick Howser replied, "Next year."

About 100 members of the Royals organization, including front office personnel and players and coaches' wives, rode in three buses to the Russell Building on the Hill, where the U.S. Army Band played, "Kansas City."

Several hundred well-wishers, many of them children, requested autographs as the Royals' party was ushered into the Senate Caucus Room for lunch with Sens. Robert Dole (R-Kan.), John Danforth (R-Mo.), Nancy Kassenbaum (R-Kan.) and Thomas Eagleton (D-Mo.).

Dole uttered probably the most appropriate words of the day after the luncheon when he said, "Obviously, we are reveling in your glory. Politicians do that from time to time.

"Some of us from Kansas," he continued, "call you the Kansas Royals. We conveniently leave out the "City." We don't understand all that I-70 stuff."

Eagleton read a Senate resolution, declaring the Royals "tenacious," among other things. And George Brett, the only player to sit at the head table, presented Royals hats to the senators.

Danforth told a story of how he became a baseball fan. Marty Marion, the famed St. Louis shortstop of the 1940s and '50s, drove into a gas station where Danforth worked and the future politician asked him to autograph his comic book. Danforth said he's been a baseball fan ever since.

"My subscription to (that comic book) has since run out," Danforth said. "But, Dick Howser, if you could get me Buddy Biancalana's autograph on a copy of the Congressional Record, I'm yours for life."

Biancalana, one of Kansas City's World Series heroes, complied with many autograph requests before the Royals gathered in the back of the room to watch selected replays of the sixth and seventh games, both of which Kansas City won to erase a 3-1 deficit.

Biancalana said he already has had contact with representatives of Johnny Carson's "Tonight Show" and David Letterman's "Late Night." But the shortstop's humorous remarks yesterday were drowned out by cheering over the replays. "You never get enough of it," Willie Wilson said. "I still get chills."