It looks like you currently now have a "trimline" type of phone with a base. There are other brands of that type.
It looks like there are some single piece (no base) phones on the market aimed at the hospital bed market.
http://www.scitecinc.com/products/healthcare/Scitec Incorporated 21105
Scitec Incorporated 21195
http://www.med-pat.com/#Med-Pat XL2050
Med-Pat Xl301
My first thought was, if you have wifi, to skip the OBi and just use an android phone with an application like CSipSimple.