Use a SNMS - Secondary Neutrals Mass Spectrometer. This instrument bombards the material with fast ions an analyses the ablated material in a mass spectrometer. Many private firms will do this process commercially.
x-ray fluorescence (XRF) spectroscopy can help you. Energy-dispersive X-ray spectroscopy (EDX) is another method which can be beneficent to quantitatively analyse the amount of doped nickel.
XRD and EDS analysis. XRD analysis is some changes in doped samples compared with pure such intensity or peaks broadening. Then EDS analysis display the how much amount of dopant content present inside the samples.
It depends on the Ni/Mn ratio you are trying to get, and on the MnO2 particle size. With low Ni and fine particles, there is the danger that most Ni will be adsorbed on the surface and you won't get much true doping (I know nothing about your preparation method of course). EDX and SNMS probably will not help you in this regard. M. Ponnar has a point - XRD might help, but only if you are attempting pretty high substitution (probably >~1%). A minor substitution will not chnage the cell parameters enough.
You may be able to use EDX or SNMS successfully, if you first develop a method of dissolving the surface layer. Then consecutive tests before and after dissolution by the same procedure will tell you - with a bit of luck - whether the Ni is mostly on the outside or uniformly distributed.
@Emanuel Cooper Thank you. I am synthesising gamma phase MnO2 in a hydrothermal synthesis by reacting a Ni salt along with Mn salt and an oxidising agent. I did perform EDS ( the doped Mn was 10% mole) but Ni was not reported. May be I should try and change the synthesis method.