Here is how an economist would approach the question with a simple model. Of course, a few assumptions are needed. Suppose aggregate production in year $t$ can be thought of as a single firm producing all output ($Y$, which is GDP) using labor ($L$), computers ($C$) and other types of capital ($K$).
$Y_{t}=f\left(K_{t},L_{t},C_{t}\right)$
Let's also make the simplifying assumption that computing speed is captured by the depreciation process of computers. If computing speed is important, then computers depreciate faster as newer, faster models come in. As a consequence, you need to invest in computers more frequently as the economy grows in order to keep the same level of output. Therefore, I argue, a sensible measure of the importance of computing power is given by the growth rate of the economy ($g$) plus depreciation rate ($\delta$), times the elasticity of output with respect to computers (that is, how many computers you need to produce an additional unit of output)
$\left(g+\delta\right)\cdot\left(\frac{\partial Y_{t}}{\partial C_{t}}\cdot\frac{C_{t}}{Y_{t}}\right)=\left(g+\delta\right)\cdot\frac{\partial\log Y_{t}}{\partial\log C_{t}}$
We can use neoclassical growth theory to obtain this measure from the data. Suppose that the economy is on what neoclassical economists call a "balanced growth path". That is: output and inputs all grow at a constant rate $g$:
$\frac{d\log Y_{t}}{dt}=\frac{d\log K_{t}}{dt}=\frac{d\log L_{t}}{dt}=\frac{d\log C_{t}}{dt}=g$
Then the ratio of yearly investment in computers as a percent of the stock of computers needs to be equal to $g+\delta$ in order for growth to remain "balanced"
$\frac{I}{C}=g+\delta$
multiply both sides by the nominal value of the computer stock ($rC$) to nominal GDP ($PY$) where $r$ is the rental rate of computers and $p$ is the GDP deflator. This gives you the following equation for the yearly nominal investment in computers, as a percentage of GDP:
$\frac{rI}{pY}=\left(g+\delta\right)\frac{rC}{pY}$
now, in equilibrium, the share of capital in total compensation $\frac{rC}{pY}$ is equal to the output/capital elasticity ($\frac{\partial\log Y}{\partial\log C}$). This means that you can rewrite the equation above as:
$\frac{rI}{pY}=\left(g+\delta\right)\frac{\partial\log Y}{\partial\log C}$
In other words, a simple neoclassical growth model suggests capital investment in computers as a percent of GDP as a measure of the "importance" of computing power in the economy.
The data you need to compute this ratio (for different countries and sectors) can be found on the website of the OECD or the EUKLEMS consortium. You'll need to search a bit for it (computers are a subgroup of "ICT capital").