British politics at the top end? It's a sea of fortysomething white male faces. The real coalition is the Coalition of Chaps, and a country that is diverse and mixed-up, is represented by a political class that's anything but. Now, though, some excellent news: at least in London we might have a real choice, because Oona King, one of the best of the Labour women, will let it be known tomorrow that she wants to be mayor.

Yes, she will be taking on the always formidable Ken Livingstone in the race to be the Labour candidate; but King, who thought that Livingstone was a good mayor, points out that he has been around for four decades. "It's nobody's birthright, and I don't believe in the hereditary principle." So there is a chance of a mixed-race woman arriving to represent this most mixed of the world's great cities.

King calls herself "culturally bilingual", with her background allowing her to see how white and black Britons have different views of the world. "When I was young, my white grandmother would say to me, if you get lost, go and find a policeman. My black family would say, run for your life if you see a policeman because he will probably beat you to death." If that sounds wild, it is worth knowing that the black side of the family lived through rough times: her black aunt lost a baby when she was eight months pregnant after being hit by a policeman during a civil rights march in the US.

Now, I know that in politics it is the quality and character of decisions that matter most, and that the "back story" comes behind these. But diversity matters at a deeper level than simply the symbolic; it just doesn't work to use young, female or black faces around a politician to send some kind of message. Real diversity is about different experiences, attitudes and instincts actually being able to reach the levers of power.

Men and women are not identical in their reactions or ways of thinking. They tend to have different experiences of going to GP surgeries, hanging round the school gate, caring for relatives, checking out streets at night, and struggling to find good part-time work. And these relate directly to how politicians' policies are shaped, and whether they work. It's not merely symbolic.

It doesn't mean that we want a simplistic checklist – one Muslim, one gay man, one northern woman – and no checklist will tell you what policies might follow. But the point of electoral democracy is to hold up a mirror to society, or at least acknowledge its variety.

And we just haven't being doing that. White, middle-aged men often fail to even notice that they talk daily to – yes, other white middle-aged men. The men who were running those recent election campaigns didn't register the fact that the entire campaign was virtually a female-free zone.

King says she wants to be a unity candidate for whites and blacks – for inner- and outer-city Londoners – and why not? She has two young children who started their lives in care, and she knows firsthand about how little support younger people get in the city: "It's about being civilised. It's about all adults taking responsibility." It's not an anti-Eton or class-war point to say that her life and experiences are closer to those of most Londoners than are Boris Johnson's. But it is a point.

We'll see how King does. She launches her campaign formally on Wednesday and deserves support from everyone who would like to see a new champion for London. This contest will run till the party conference in the autumn, just like that other contest for the leadership of the Labour party itself.

Here too we have had a pretty much white, boys-only contest. Yes, it's good to see Diane Abbott having a go, but she is going to struggle to get past the nominations hurdle – unless John McDonnell does the decent thing and steps back to allow a woman to get through to the contest. But the biggest disappointment is that not a single one of the former female cabinet ministers has entered the fray.

Individual reasons vary, but I can sum them up like this. Either (so far) they have decided that they are too old, and it's time to give the next generation a chance; or they are still too embroiled with children and therefore not ready. But let's think this through. It means that only a fortysomething childless woman would be a possible female candidate. That's crazy.

One idea that is being muttered about, and which many people will also call crazy, is that two female candidates could stand jointly, as a job share. Tessa Jowell and Yvette Cooper, for instance?

Before you hoot with derision, just remember how many other things were once dismissed as crazy, from job sharing in general, to women being allowed to vote and stand for election to the Commons. And after all, with Clegg-Cameron we already have a prime ministership which is pretty close to a job share (not an equal one, I concede). Almost every change in politics starts with wiseacres saying: "You can't possibly do that."

It's at least worth thinking about and it would do Labour no harm to spread the contest out more widely. Its inquest and its debate about future direction will simply be more meagre and less convincing if it lacks a major input from senior female politicians. On immigration, housing, health and education, we need to hear their voices too.

In the end, Labour may well vote for a Milibrother, and they are undoubtedly two of the brightest and best. These are still early days, and every extension of the field improves it. It was good to hear Andy Burnham with a northern, working class voice, make his pitch. It was good to begin to hear an honest exchange about the Iraq war. I like the way the "Gordon Brown factor" has been acknowledged, but with a certain dignity and without jeering or cruelty. In general, we should have high hopes for this contest. It could turn out to have that quality Labour finds so hard to live up to – brotherhood.

But of course that's my point. Labour is a party of sisterhood as well as brotherhood, and will never recover unless it roots itself once more in the experiences of ordinary women and men. That cannot be done by little posies of female faces around "real" male leaders, or by recruiting a few female research assistants or by focus groups. It needs politicians with varied life experiences, not all from the same age group, class and gender, who are listening intently, with their whole selves. I think King can do this in London. Who will help do it for Britain?

Written by Jackie Ashley and published in The Guardian on 24 May 2010