Colleges have warned that they face losing more than ВЈ170m because of delays in getting funding for refurbishment.
Meanwhile Atkins has warned that hundreds of construction jobs could go if the programme is not brought back on track.
Government trouble-shooter Sir Andrew Foster is conducting an urgent review of the situation.
Mr Haysom said: "I don't need to wait for that report to be published before making my decision because it will, I'm sure, confirm what I now know - that there have been failures in the way that the LSC has managed the programme.
"No matter where those mistakes have been made and no matter how many people have been involved in the capital programme, as the chief executive of the LSC I am, of course, finally accountable. That is why I have made my decision to step down at this time."
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