That she very soon went back to Dorking

There was an Old Man who supposed (был /однажды/ старик, который полагал),

That the street door was partially closed (что дверь на улицу частично закрыта = прикрыта; partially — немного, отчасти, частично, частью);

But some very large rats (но несколько очень крупных крыс),

Ate his coats and his hats (съели его пальто и его шляпы; eat — кушать),

While that futile old gentleman dozed (пока этот несерьезный/поверхностный старый джентльмен дремал).

suppose [sə'pəuz], partially ['pɑ:ʃ(ə)lɪ], futile ['fju:taɪl]

There was an Old Man who supposed,