  Recently, Contreras et al. \cite{Contreras:2021yxe} introduced a new type of
black hole, called hairy Kerr black hole (HKBH), which describes a Kerr BH
surrounded by an axially symmetric fluid with conserved energy momentum tensor.
In this paper, we compute the gravitational waves emitted from the extreme mass
ratio inspirals around the HKBHs. We solve the Dudley-Finley equation, which
describes the gravitational perturbations of the HKBH, and obtain the energy
fluxes induced by a stellar-mass compact object moving on the equatorial,
circular orbits. Using the adiabatic approximation, we evolved the radii of the
circular orbits by taking into account the backreaction of gravitational
radiation. Then we calculate the dephasing and mismatch of the EMRI waveforms
from the HKBH and Kerr BH to assess the difference between them. The results
demonstrate that the EMRI waveforms from the HKBH with deviation parameter
larger than $0.001$ and hair charge smaller than $1.5M$ can be discerned by
LISA.
