Martin Luther King - Michael King Junior before God - died on 4 April 1968; shot on the second-floor balcony of a Memphis motel. In a speech the day before he told a crowd that death no longer concerned him. "I've been to the mountaintop," he said. "I've seen the promised land."

Katori Hall's play, which pipped Jez Butterworth's Jerusalem to an Olivier Award in 2009, is set that night in Room 306 of the Lorraine Hotel. It shows us King behind closed doors - and Gbolahan Obisesan looks the spit of the man. Ready to knock up another speech, he calls room service and opens the door to Camae (Ronke Adekoluejo), an African-American maid on her first day in the job. She's a flirtatious streak, a militant edge and a seemingly endless supply of Pall Mall cigarettes – by miraculous coincidence, King's exact brand. Not such a coincidence, as it turns out. Camae's a godsend: an angel come to ready King for death.

Theirs is a meeting of minds, bodies and, ultimately, souls. They're two people connected by a cause, but from very different worlds, fighting the same fight in very different ways; privileged, educated King with pacificism, and street-sharp Camae with the Black Panthers. It makes for a feisty clash - assimilation against positive segregation, peaceful means against aggressing - that's always underpinned by attraction. They circle each other like planets, pulled together; two bodies always aware of each other. It's impossibly sexy; intellectual stimulation that crackles.

Roy Alexander Weise, winner of this year's JMK Award for emerging directors, ups the supernatural element. Rather than the odd flicker of proof - cigarettes that self-light, flowers that spring from tears through the carpet - he lets Rajha Shakiry's beige Sixties suite flood with colour and light. It looks divine, but there are gains and losses. Though it ups the surreality of their encounter, it can make Hall's play look a touch naff in places. There's magic enough in King talking to God on a lowly motel phone; add a pink glow and angelic underscoring, and it's overkill. It lessens the leap of faith involved.

At the same time, however, it frees up the play, building to the most extraordinary climax: ten breathtaking minutes of theatre. With a kiss, Camae shows King the world's future. Superb throughout, Adekoluejo speeds off through a syncopated beat-poem rushing through half a century. Nina Dunn's videos light the walls like newsreel. She dances the future - and the future is that same fight; good and bad, OJ Simpson next to Obama. It ends not with words, but with a gesture: her hands up above her head.

In 2009, The Mountaintop asked a question. With a black president in place, Hall's play questioned whether the summit had been reached. It warned against complacency. It's different today, seven years on. Against the backdrop of Black Lives Matter, it's a call to action backed up by history. It's as empowering as it is powerful.

In humanising King, Hall shows us the man beneath the icon: one whose feet smell, who jumps at thunder, who lusts and cheats on his wife. Obisesan captures the contradictions and if, at first, that seems to undercut this cultural hero, suggesting that we've sanctified a man like any other, Hall turns the thought back on itself.

King achieved all he did as a man. He wasn't predestined to do so, nor did he start out or speak up as the idol we think of today. He was a man; one who feared death, who wrestled with uncertainty and conscience, who gave up a quiet life for a cause. The Mountaintop shows his courage anew and, what's more, it encourages it in us. With the same courage and conviction, any of us could step up now as he did then. "Why me?" King asks of God. "Why not?"

The Mountaintop runs at the Young Vic until the 29 October.