I am from Beijing and I have been to all the places you mentioned.
There must be rental houses in the community, called B&Bs. 50 yuan a day is enough, but if it is a hotel, the price of 148,168 yuan for a single room is more common.
The Forbidden City is definitely a place to go, and Houhai and Beihai are also good. There are a lot of foreigners there, and it feels like old Beijing. The Bell Tower is very close to Houhai, so you can visit it in one day. Go out to Di'anmen West Street, and cross the road to Qianhai and Houhai. In Houhai, go eat that one (the 9-door snack is really not that delicious) and further north is Yandai Xiejie. You may also know that there is a very blocked branch under the Gulou Tower. If you continue walking east on Gulou East Street, you can reach the north entrance of Nanluogu Lane. You can also go south to Di'anmenwai Street, pass through Maoer Hutong, or go to Nanluogu Lane.
The 798 Art District is not far from my home. To tell you sincerely, it is quite average. I recommend not going here.
I don’t know if you have a large-scale amusement park there. There is Happy Valley in Beijing, which has the Sun Cart (very scary), exciting games like Two Heroes of Heaven and Earth, and there is also Ant Kingdom, which is suitable for the timid. People play. You can get there by taking Metro Line 10.
The cost in Beijing is not very high. The important thing is to see what you eat and live in. A slightly mid-range price of 5 or 60 a day is more than enough (for meals). Staying in a B&B at night is about 50. Beihai The entrance fee is 15 yuan. There is no entrance fee for Houhai. The entrance fee for Happy Valley is 80 yuan, which is a pass.
It is quite suitable to stay for 5 or 6 days.