The Wizards parted with two seldom-used players in forward Jan Vesely and point guard Eric Maynor. Vesely goes to Denver while Mayor goes to Philadelphia, which also received second-round picks from both the Nuggets and Wizards.
Vesely and Brooks, who was acquired in exchange for shooting guard Jordan Hamilton, are expected to join the Nuggets for their game at Chicago on Friday night.
Miller's departure ends a 50-day saga in which he hadn't played a single minute for Denver even after Ty Lawson (rib) and Nate Robinson (torn ACL) were sidelined as the Nuggets nosedived into the All-Star break on a four-game skid.
Miller got into a shouting match with first-year coach Brian Shaw in the waning minutes of a loss to the 76ers on New Year's Day. The Nuggets initially suspended Miller for two games, then rescinded the suspension and announced that he would instead be excused from all team activities for personal reasons.
The two were never able to set aside their differences, so general manager Tim Connelly worked out a deal Thursday that Shaw said he thought was a relief "for both sides."
"You know, nobody wanted to have the situation come down to what it came to, so we wish Andre the best in Washington. It gives him a chance to do what he loves to do and get out there and play basketball, so it's a better situation for him," Shaw said before the short-handed Nuggets' game at Milwaukee.
Miller, who turns 38 next month, has played with five teams over 15 NBA seasons. He averaged 5.9 points and 3.3 assists for the Nuggets this season.
Vesely, who's just shy of 7 feet, was the sixth overall pick of the 2011 draft. In two-plus NBA seasons, he's averaged 3.5 points and 3.4 rebounds in 15.2 minutes.
Brooks was selected 26th overall by Houston in 2007 and had a breakout playoff run in 2009, averaging 16.8 points in 13 games, including a 34-point performance in Game 4 of Houston's second-round series against the Lakers. He then averaged 19.6 points and 5.4 assists, both career bests, in 2009-10, when he was named the league's most improved player.
In 43 games with Houston this season, Brooks averaged 7.0 points and 1.9 assists in 16.7 minutes. He had to agree to the trade, and he did so.
Brooks figures to play many more minutes for the Nuggets even when Lawson returns to the lineup.
The moves give the Nuggets future flexibility because both Brooks and Vesely have expiring contracts.
The Nuggets had just nine players available against the Bucks.
"But we've been kind of operating that way the last couple of weeks, so I think the guys that are here are accustomed to that," Shaw said. "That's just opportunity for other guys to get in there and play and make the most of it."
Coaching short-handed has "been a challenge, but it's been fun," Shaw said. "I get to see how creative I can be in certain situations. You go from having a semi-healthy roster and basically three point guards to having none, we've had to juggle some things around. It's all part of the job, and for me, I've gotten probably 10 years of experience in this first half of the season."
Also Thursday, the Nuggets announced that center JaVale McGee had surgery in West Palm Beach, Fla., to repair a stress fracture in his left tibia, officially ending his season. The Nuggets' $10 million-a-year center played in just five games this season, averaging 7.0 points and 3.4 rebounds.
"I'm glad that he was finally able to try something that hopefully will get this injury to start to heal," Shaw said.
AP freelancer Chris Jenkins contributed from Milwaukee.
Follow AP Sports Writer Arnie Melendrez Stapleton on Twitter: http://twitter.com/arniestapleton