For the location of your grandmothers home hopefully the municipality of Apeldoorn can help you further. Their website is in Dutch only, nevertheless you can fill in this contact form. https://fd10.formdesk.com/apeldoorn/contactformulier_apeldoorn Onderwerp = subject > select “overige vragen” , Naam = name, Vraag = question and red button "Naar bevestiging>>" is send.
Apeldoorn is not blessed with an attractive historic city centre. The main draw is as Mary remarks the Loo Palace or Paleis Het Loo in Dutch. Once a Royal Palace it’s packed with history, so always interesting to my idea. However the main building is undergoing renovation and according plan the Palace will open it’s doors again coming April the 15th. So if nothing changes May will be perfect for a visit. Further if of interest, to Dieren runs a historic railway line for steam trains, museum depot in Beeksbergen.
If you look for a quaint old town (to visit or to stay) I can highly recommend nearby Zutphen or Deventer, the first has one of the best preserved historic city centres of the Netherlands. Both places idyllacilly located along the Yssel river, you won’t see much international tourists there, besides a handful of Germans. From Apeldoorn Zutphen is within 20 minutes reach by train, Deventer about half that time.
If the Kröller-Müller Museum with it’s famous Van Gogh collection is of interest you can take the bus from Apeldoorn to nearby Otterlo and change there to the bus to the museum. For the Open Air Museum near Arnhem you can take the train, from the railway station there the bus to the entrance. For the Airborn Museum (Operation Market Garden) in Oosterbeek just east of Arnhem you can take the train too. From Zutphen shortest travel time to both museums.
Edited: The Netherlands is now in lockdown, nevertheless according the latest info restrictions will be soon less, but as opening museums have a lower priority they remain closed for the moment and hard to say for how long!