so has the patient not sat up in the 6 months since the #?

If she hasn't sat up and has advanced contractures I think you may be being a bit optimisitic re: transfers, particularly with a banana board as this requires quite a lot of motor skill. Would she have the dynamic balance and strength to do this? As for the contractures the evidence on stretching is that once contractures have occurred the stimulus isn't significant enough to reverse them, so not much you really can do.

Serial casting for the ankles and knees is a consideration although this is a prolonged treatment, can be very uncomfortable with skin pressure and friction risks. Also I would be cautious doing this with a patient with dementia - you need cooperation throughout the day and patient needs to understand what is happening. Serial casting is best done if you have some experience with it - it is quite an art.

However why don't you start on sitting unsupported as a goal and possibly lying to sitting is she is up to it?

Another thing you could try, for the purpose of just getting her upright and morale - is put her on a tilt table - depending on how bad the plantar flexors and knees are contracted - you could arrange a raised heel so that she is weightbearing through the hind foot.