You could take the Westbound Piccadilly line to Green Park and change for the Jubilee Line to Westminster. This will probably be quicker than waiting for a Circle line.
Although green park is not a 'step free' station:
"At Green Park you can only change step-free between the Piccadilly and Jubilee lines. You cannot change step-free with the Victoria line, and you cannot exit or enter Green Park step-free either."
When alighting from a Westbound Piccadilly line train, there is a 151mm step, and a 0mm gap.
When boarding an Southbound Jubilee line train, there is a 180mm step, and a 0mm gap.
Please be aware that if travelling via Green Park, if you're at Green Park and it is evacuated, there is no step-free access to street level.
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Kings Cross St Pancras
Lift to Piccadilly Line (Westbound)
Alight at Green Park
Step-Free interchange between Piccadilly & Jubilee lines at Green Park
Board the Jubilee Line (Southbound)
Alight at Westminster
Lift to Street Level
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