We focus on the task of finding a 3D conductivity structure for the DO-18 and DO-27 kimberlites, historically known as the Tli Kwi Cho (TKC) kimberlite complex in the Northwest Territories, Canada. Two airborne electromagnetic (EM) surveys are analyzed: a frequency-domain DIGHEM and a time-domain VTEM survey. Airborne time-domain data at TKC are particularly challenging because of the negative values that exist even at the earliest time channels. Heretofore, such data have not been inverted in three dimensions. In our analysis, we start by inverting frequency-domain data and positive VTEM data with a laterally constrained 1D inversion. This is important for assessing the noise levels associated with the data and for estimating the general conductivity structure. The analysis is then extended to a 3D inversion with our most recent optimized and parallelized inversion codes. We first address the issue about whether the conductivity anomaly is due to a shallow flat-lying conductor (associated with the lake bottom) or a vertical conductive pipe; we conclude that it is the latter. Both data sets are then cooperatively inverted to obtain a consistent 3D conductivity model for TKC that can be used for geologic interpretation. The conductivity model is then jointly interpreted with the density and magnetic susceptibility models from a previous paper. The addition of conductivity enriches the interpretation made with the potential fields in characterizing several distinct petrophysical kimberlite units. The final conductivity model also helps better define the lateral extent and upper boundary of the kimberlite pipes. This conductivity model is a crucial component of the follow-up paper in which our colleagues invert the airborne EM data to recover the time-dependent chargeability that further advances our geologic interpretation.