Here is an example with the query in the body. My queries are quite complex and have a lot of let statements and therefore passing the query in the body is easier. There are some PowerShell quirks in the example below, but I'll update with a C# example tomorrow.
The let statement in the example below is pretty pointless, it's mostly there to show that you can do complex queries with let expressions etc.
AppId is the Application Insights resource ID - and NOT the instrumentation key. The API key is just a long string and you can create up to 10 of them AFAIK.
You will find both the id and keys under API Access (I've added a screenshot as it's easy to get them confused). When you use the app() function use the app id.

$app1Id = "GUID"
$app2Id = "GUID"
$app1Key = "string"
$app2Key = "string"
# EXAMPLE: "X-Api-Key" = "key1:GUID1,key2:GUID2"
$headers = @{ "X-Api-Key" = "${app1Key}:$app1Id,${app2Key}:$app2Id"; "Content-Type" = "application/json" }
# EXAMPLE: "query" = "union app('GUID1').something, app('GUID2').something | limit 5"
$query = @{"query" = "let days=1d;union app('$app1Id').exceptions,app('$app2Id').exceptions | where timestamp > ago(days)"}
$body = ConvertTo-Json $query | % { [regex]::Unescape($_) }
$result = Invoke-RestMethod "https://api.applicationinsights.io/v1/apps/$app1Id/query" -H $headers -Body $body -Method POST
The query above will return all the exceptions for the two Application insights resources for the last day. You can do a query across 10 resources at the time of writing, 200 requests per 30 seconds or a max of 86,400 requests per day (UTC). Other limits apply if you use ADD.
NOTE: the extra {} in the header is a PowerShell quirk in regards to variables and the use of the colon char, and as you can see in the example you should not bracket the keys in the header :)