Ahh I feel your pain from visiting my dh mid week (at his hotel) with just my 1yo-2yo... It is really tough to find things to do in a town you're not 'normally' in and then to occupy them out the entire day... and when it's raining or cold :( :( It was really hard!
Top tip: Make up a bag/backpack of snacks/water, travel games, crayons, coloring, a small 'hand-sized' football, and maybe sand/digging toys. Take it with you always and keep it well stocked...let the kids pick out some of the snacks/treats at the store with you (as this too will take time out of your day). Also, bring a book, magazine or newspaper for yourself so that when you're not needed or wanted as much (at the park or where-ever) so can occupy yourself a bit without getting bored.
Also, I let your kids take turning picking where they want to go/what they want to do.
During nice weather:
park in the morning with a picnic blanket, Lunch on picnic blanket or as a 'treat' in cafe or at the occasional fast food joint. Shopping, museums, zoos, wildlife parks, woods, pools or another play park in the afternoon.
Poor weather:
waterproofs and park, cafe for lunch, afternoon indoor play, shopping, museums, library,pool, waterproofs and park or zoos/wildlife parks...
Nasty weather:
Shopping (we were regulars at Pets at Home)
, museums, library,pool, indoor play...
Even so we never were very comfortable during the truly bad weather and often times we'd find ourselves back at the hotel early watching a movie on the computer... (can you manage this somehow?) And it's very expensive actually 'hanging' out all day long... money to get in places, drinks/snacks to be had...
I often times would stay at a quiet cafe or restaurant for a good long time playing with dd at the table (making sure to buy drinks/cake or have a meal and to pack up and go if it got busy).
And have a reasonable chat with you ex and ask if maybe you can swap, and she go out for the day or morning or afternoon if the weather is poor... and ask for her ideas of what and where you can go.