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I have tried to do dimensional analysis on the equation $v=u+at$. It has resulted in $v=2\,\mathrm{ms}^{-1}$. However, the units of velocity are clearly just $\mathrm{ms}^{-1}$. What have I done wrong?

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Dimensional analysis just looks at the dimensions. So adding something with units of $\rm{m\cdot s^{-1}}$ to something else with units of $\rm{m\cdot s^{-1}}$ doesn't result in a new unit of $2\rm{m\cdot s^{-1}}$. When adding two things with the same unit you get the same unit back (which is that only way you can add two numbers by the way).

It's like if you travel $5$ meters and then travel $4$ meters in a single direction: You travel a total of $5\ \rm m + 4\ \rm m=9\ \rm m$, not a total of $9\ 2\rm m$

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Dimensional analysis uses a slightly different kind of addition, namely $m + m = m$. What this means is that adding a length in meters to a length in meters results in a length in meters. Multiplications by unitless quantities (the 2 in your example) are ignored.

The purpose of dimensional analysis is to check that calculated quantities make sense, that addition and subtraction are only between like units, and thay the result of a calculation has correct units. So, the math needed is simpler than for the numerical calculations.

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