It's a hard question to define but, essentially, are there enough two-storey houses in the suburbs to balance out all the flats in the centre?
Now, finally, I've found some actual data to help me come up with an answer. It comes courtesy of James Gleeson's report Housing in four world cities: London, New York, Paris and Tokyo which he published for the GLA this week. The report is here, Jim's carefully-selected summary-tweets are here and a quick spoiler would be "Compared to the other three cities, London’s housing is older, lower-rise, more likely to be social housing and less likely to be vacant." It's a fascinating comparative read. Best of all, Jim's full set of data has been provided in a downloadable spreadsheet, and I've been able to use that.
Here's a good start...
Type
Number of homes
%
House or bungalow 
1,732,230
51%
Low rise flat
1,419,989
41%
High rise flat
277,446
8%
Total
3,429,665
 100%
Let's be clear what this data is. It's the number of homes in London, not the number of buildings, nor the number of people. There are about 3½ million homes in London. Just over half of them are houses or bungalows. Just under half of them are flats. The data is from 2016.
That's interesting, but to clarify the average height of a home we need data about storeys instead...
Storeys
Number of homes
%
1
63,395
2%
2
1,490,434
43%
3
981,658
29%
4
433,668
13%
5
183,064
8%
6 or more 
277,446
5%
Total
 3,429,665
 100%
Let's be clear what this data is. It's the number of homes in buildings with that number of storeys, not the number of buildings, nor which floor people live on. 43% of London's homes are in two-storey buildings. Three-quarters of London's homes are in buildings with one, two or three storeys. Only 5% of London's homes are in buildings over 5 storeys high.
So, how many storeys high is the average London home?
There are three types of average, two of them easier to calculate than the other.
The mode is two-storey buildings, because that's by far the most frequent (43%).
The median is three-storey buildings, because if you lined up all of London's residential buildings in height order, the middle one would have three storeys.
And the mean is more complicated.
To find the mean you add up all the storeys and then divide by how many homes there are. For homes in buildings up to five storeys that's easy because we know how many there are. The problem is with "6 or more", because we don't know how high the tallest blocks of flats are.
If we assume the minimum case, i.e. all highrise flats have six storeys, this gives an average of almost precisely 3. If we assume all highrise flats have seven storeys, this only knocks the mean up to 3.1. We'd have to assume all highrise flats had 13 storeys before the average rounded up to 4 rather than down to 3. And 13 storeys is definitely pushing it, which leads me to conclude that the mean is 'three and a bit', and nearer to 3 than 4.
So, the average London home is in a building with two storeys (mode) or three storeys (median, mean).
But in Paris and Tokyo it's roughly five.
And in New York it's more like seven.
London's a lot more low-rise than these other world cities.
This table may help to explain why...
Built
Number of homes
%
Before 1919
924556
27%
1919 to 1944
1,030,945
30%
1945 to 1964
449,437
13%
1965 to 1980
422,008
12%
1981 to 1990
152,826
5%
Post-1990
449,893
13%
Total
3,429,665
 100%
And so might this...
This is a map of population density, which shows that some parts of inner London are quite crowded but most of the outer suburbs really aren't. Kensington, Islington and the East End might be densely packed, but that's not true of Hillingdon, Havering and Bromley. A heck of a lot of London is pre-war houses with gardens on residential streets, and not so much is post-war flats.
And that's why the average London home is two, maybe three, storeys high.
(For some less woolly statistics and conclusions, check out @geographyjim's proper GLA report)