Measuring production of $W$-boson pairs at particle colliders provides an important test of the predictions of Standard Model (SM) of particle physics in both perturbative quantum chromodynamics and electroweak domains. In this measurement, fiducial and differential cross-sections are obtained using data recorded with the ATLAS detector between 2015 and 2018 in proton-proton collisions at center-of-mass energies of 13 TeV at the Large Hadron Collider, corresponding to an integrated luminosity of 140 fb$^{−1}$. The number of events due to top-quark pair production, the largest background, is reduced by rejecting events containing jets with $b$-hadron decays. Background contributions from top-quark and non-prompt leptons are estimated using data-driven techniques. In contrast to most of the previous measurements that enhance the $WW$ signal purity by vetoing hadronic jets in the final state, the first measurement of $WW$ cross-sections using a fully jet-inclusive selection is presented in this work. The fiducial $WW$ cross-section is determined in a maximum-likelihood fit with an uncertainty of 3.1%, providing the most precise cross-sections of $WW$ production achieved in hadron-hadron collisions to date. The measurement is extrapolated to the full phase space, resulting in a total $WW$ production cross-section of $124 \pm 4\ \mathrm{pb}$. Differential cross-sections are measured as a function of twelve observables describing the kinematics of the $WW$ system. State-of-the-art theory predictions of the SM are in excellent agreement with the data.