West Ham do not appear to be finished there, though, with talks said to be ongoing with Felipe Anderson and Fabian Balbuena, while a £25m bid for Newcastle United skipper Jamaal Lascelles has been knocked back.
Sullivan hopes to have further good news soon, telling the club's official website: "We are delighted to have secured two very exciting signings in the space of three days. As a board, we assured our supporters that we would be looking to bring in high-calibre players, and Jack and Andriy fit that bill perfectly.
"Both of them have played at the highest level and we look forward to seeing that quality on show on a regular basis in the Claret & Blue. Jack also knows the Premier League very well, and I know all our supporters will join me in welcoming them both to the London Stadium.
"They become our fourth and fifth new signings of an already busy summer, after the additions of Ryan Fredericks, Issa Diop and Lukasz Fabianski. We also hope to have more exciting news regarding incomings in the near future as we aim to provide Manuel with the strongest squad possible."
With Wilshere and Fredericks joining on free contracts, United are understood to have splashed out £46.5m on the other three players.