Space AI has become increasingly important and sometimes even necessary for government, businesses, and society. An active research topic under this mission is integrating federated learning (FL) with satellite communications (SatCom) so that numerous low Earth orbit (LEO) satellites can collaboratively train a machine learning model. However, the special communication environment of SatCom leads to a very slow FL training process up to days and weeks. This paper proposes NomaFedHAP, a novel FL-SatCom approach tailored to LEO satellites, that (1) utilizes high-altitude platforms (HAPs) as distributed parameter servers (PSs) to enhance satellite visibility, and (2) introduces non-orthogonal multiple access (NOMA) into LEO to enable fast and bandwidth-efficient model transmissions. In addition, NomaFedHAP includes (3) a new communication topology that exploits HAPs to bridge satellites among different orbits to mitigate the Doppler shift, and (4) a new FL model aggregation scheme that optimally balances models between different orbits and shells. Moreover, we (5) derive a closed-form expression of the outage probability for satellites in near and far shells, as well as for the entire system. Our extensive simulations have validated the mathematical analysis and demonstrated the superior performance of NomaFedHAP in achieving fast and efficient FL model convergence with high accuracy as compared to the state-of-the-art.