The New Horizons flyby of the cold classical Kuiper Belt object 486958 Arrokoth (2014 MU69) showed it to be a contact binary. The existence of other contact binaries in the 1-10km range raises the question of how common these bodies are and how they evolved into contact. Here we consider that the lobes of Arrokoth formed as a binary in the Solar nebula, and calculate its orbital evolution in the presence of gas. We find that the sub-Keplerian wind of the disk brings the drag timescales for 10km bodies under 1 Myr for quadratic-velocity drag. In the Kuiper belt, however, the drag is linear with velocity and the effect of the wind cancels out as the angular momentum gained in half an orbit is lost in the other half: the drag timescales for 10 km bodies remain over 10 Myr. In this situation we find that a combination of drag and Kozai-Lidov oscillations is a promising channel for collapse. We analytically solve the hierarchical three-body problem with drag and implement it into a Kozai plus tidal friction model. The permanent quadrupoles of the lobes make the Kozai oscillations stochastic, and as drag shrinks the semimajor axis it more easily allows the fluctuations to bring the system into contact. Evolution to contact happens very rapidly (with 104 yrs) in the pure, double-average quadrupole, Kozai region between ~85-95°, and within 3 Myr in the drag-assisted region beyond it. The synergy between J2 and gas drag widens the window of contact to 80-100° initial inclination, over a larger range of semimajor axes. As such, the model predicts an initial contact binary fraction of about 10% for the cold classicals in the Kuiper belt. The speed at contact deviates from the escape velocity only because of the oblateness. For Arrokoth, the contact velocity should be 3.3–4.2 m/s, in line with the observational evidence from the lack of deformation features and estimate of the tensile strength.