Can anyone identify this plant, please?

Hi there,

I have had this plant on my window sill now for quite some time.  Unfortunately I have managed to lose the identification card. 
I think it's a Yucca - but I am having difficulty in identifying the particular type myself on the internet.
I have re-potted it twice... it continues to grow upward on this thin spindly stem. It has lost its lower leaves. And some are browning. I'm not entirely sure how best to care for it. 
 
It has grown too large for the window sill haha.. so it needs to be potted and moved. 
Please see the attached picture..

Firstly, can anyone tell me what the plant is?
Also, shall I re-pot in a taller pot?  It undoubtedly needs a support stick... as it also leans.
Will the lower leaves ever grow back? Will the stem thicken?
It's growing so top heavy on a thin stem that it needs support.
Any advice will be gratefully received ☺

Thanks ever so much!