This paper introduces a novel framework aimed at designing integrated waveforms for robust integrated sensing and communication (ISAC) systems. The system model consists of a multiple-input multiple-output (MIMO) base station that simultaneously serves communication user equipments (UEs) and detects multiple targets using a shared-antenna deployment scenario. By leveraging Kullback-Leibler divergence (KLD) to holistically characterise both communication and sensing subsystems, three optimisation problems are formulated: (i) radar waveform KLD maximisation under communication constraints, (ii) communication waveform KLD maximisation subject to radar KLD requirements, and (iii) an integrated waveform KLD-based optimisation for ISAC that jointly balances both subsystems. The first two problems are solved using a projected gradient method with adaptive penalties for the radar waveforms and a gradient-assisted interior point method (IPM) for the communication waveforms. The third, integrated waveform optimisation approach adopts an alternating direction method of multipliers (ADMM) framework to unify radar and communication waveform designs into a single integrated optimisation, thereby synergising sensing and communication objectives and achieving higher overall performance than either radar- or communication-only techniques. Unlike most existing ISAC waveform designs that regard communication signals solely as interference for sensing, the proposed framework utilises the holistic ISAC waveform-that is, the superimposed communication and sensing signals-to boost detection performance in the radar subsystem. Simulation results show significant improvements in both radar detection and communication reliability compared with conventional zero-forcing beamforming, identity-covariance radar baselines, and traditional optimisation approaches,..