I just solved this problem (DSi working on a secured network WAP/ WAP2):
Step 1: You need to make sure your AIRPORT or TimeCapsule wireless router is setup to support N,B,G protocols:
1._ Login to your Airport Console
2._ Under Airport, go to Wireless
3._ Under the Radio Mode you need to select 802.11n (802.11b/g compatible)
4._ Under Security you need to select WPA/ WPA2 Personal
Step 2: Copy your DNS server being used by your Airport router
1._ In the Airport Console, click on Internet
2._ Under DNS servers there are 2 entries. Copy them. The first entry is your Primary DNS server and the second Entry is your secondary DNS server
Step 3: Configure Wireless in your DSi
1._ Once you have power your DSi, Go to System Settings
2._ Click the BlueArrow twice to get to the Internet option
3._ Click on Internet
4._ Click on Connection Settings
5._ Click on Advance Setup
6._ Click on Connection 4 or any available (I will recommend to start with a fresh none use connection)
7._ Click Search for an Access Point
8._ Choose your wireless Network name
9._ Enter your security key. Notice, it will recognized the protocol as WPA2-PSK(TKIP)
10. Click OK
11._ Click OK again to Save the settings
12._Click OK on Begin connection test - Note: You will get an error like Error Code 052111-1. Don't worry yet
13._ Click back
14._ Click connection setting, click Advanced setup, Click setup complete on the connection number you just created before, Click on Change Settings
15._ Click on The Blue arrow 3 times until you see the Auto-Obtain DNS. CLICK NO. Click Detail Setup. Enter the 2 numbers you copies step 2.2. and click OK on each entry. Then OK again.
16._ Click Save
17._ Now the Big Moment. It will ask you to begin a connection test. Click Yes.
You will get a message that your connection test was successful.
Now enjoy your DSi...