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I am using the usual estimator for kurtosis, $\hat{K}=\frac{\hat{\mu}_4}{\hat{\sigma}^4}$, but I notice that even small 'outliers' in my empirical distribution, i.e. small peaks far from the center, affect it tremendously. Is there a kurtosis estimator which is more robust? Thanks.

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There are several. You will find an exhaustive comparison in this link to an ungated version of the paper (proper reference at the bottom of this answer).

Because of the constraints of the problem, the breakdown of the most robust of these algorithms (the L/RMC) is at most 12.5%. An advantage to the the L/RMC is that it is based on quantiles and remain interpretable even when the underlying distribution has no moments. Another advantage is that it does not assume symmetry of the distribution of the uncontaminated part of the data to measure tail weight: in fact, the algorithm returns two numbers: the RMC for the right tail weight and the LMC for the left tail weight.

The robustness of an estimator can be measured by its breakdown point. However, the notion of breakdown point is a complicated one in this context. Intuitively, it means that an adversary would need to control at least 12.5% of your sample to make this estimator take on arbitrary values (that is to be understood as an arbitrary value within the range of values that the estimator can return, since the measure of tail weight is always in $[0,1]$ by construction: no amount of contamination can for example cause the algorithm to return -1!). In practice, one finds that one can replace about 5% of the sample with even very pathological outliers without causing the most affected of the estimates (there are always two) to depart too much from the value it had on the uncontaminated sample.

The L/RMC is also widely implemented. For example you can find an R implementation here. As explained in the article linked above, to compute the L/RMC, you need to compute the MC (the estimator implemented in the link) separately on the left and right half of you data. Here, (left) right half are the sub-samples formed of the observation (smaller) larger than the median of your original sample.

  • Brys, Hubert, Struyf. (2006). Robust Measures of Tail Weight.
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