I would have to presume that the problem stems from the data appearing as an actual string to the sorting algorithm, thus producing lexicographical sorting (e.g. sorting as a string). You have plenty of choices to choose from if you want to introduce easy sorting: AngularJS has an orderBy directive, jQuery has a tableSort, module you can add into it (and others, just search for them), as well as most other JS frameworks. Of course, it would be fairly easy to roll your own sorting command in about 20-40 lines of code, as outlined on StackExchange.

Internet Explorer 7 and 8 are shortly losing their support in salesforce.com, and it is generally time to upgrade as developers as a whole work towards supporting standards instead of the browser/feature detection fragmented past spawned by the browser wars. Make sure you're testing your code in IE 9 or, even better, IE 10, which has already dwarfed previous versions, largely thanks to the automatic upgrades provided by Microsoft.

Personally, I would write my own table sorting algorithm using a data attribute or JavaScript function parameter on the header column that would instruct my function to treat the values as either text, numbers, or dates, and parse the values in each column appropriately with some handy regular expressions to strip numbers down to something I could pass into parseFloat, sort the resulting set of data, then re-order the tables. I don't think this would weigh in much over 100 lines of code, tops, given the links I provided above.

Edit

Here's an updated link for the code provided below.

In Lightning, there's no need to import a library or call out to the server, you can just write your own. Note: I've not tested this, so you may need to tweak it to suit your needs.

({
    sortByName: function(component, event, helper) {
        var currentOrder = component.get("v.sortAsc"),
            currentList = component.get("v.records");
        currentOrder = !currentOrder;
        currentList.sort(function(a,b) {
            var t1 = a.Name == b.Name, t2 = a.Name < b.Name;
            return t1? 0: (currentOrder?-1:1)*(t2?1:-1);
        });
        component.set("v.sortAsc", currentOrder);
        component.set("v.records", currentList);
    }
})

Your actual click handler needs a minor adjustment as well:

onclick="{!c.sortByName}"

This code assumes two variables, namely sortAsc, a Boolean, and records, a list of Opportunity records.

To make it more palatable for multiple fields to sort by, you'll want to put this logic into a helper function, then you can replace all the .Name references with a dynamic attribute, like a[field] < b[field].

I actually wrote a demo for this. It's not "pretty", but it demonstrates the functionality:

OpportunityHelper.cls

public class OpportunityHelper {
    @AuraEnabled public static Opportunity[] loadOpportunityRecords() {
        return [SELECT Name, Amount FROM Opportunity WHERE Amount <> NULL LIMIT 10];
    }
}

opportunity.app

<aura:application controller="OpportunityHelper">
    <aura:attribute type="Opportunity[]" name="records" />
    <aura:attribute type="Boolean" name="sortAsc" />
    <aura:attribute type="String" name="sortField" />
    <aura:handler name="init" value="{!this}" action="{!c.doInit}" />

    <table>
        <thead>
            <td onclick="{!c.sortByName}">Name</td>
            <td onclick="{!c.sortByAmount}">Amount</td>
        </thead>
        <tbody>
            <aura:iteration items="{!v.records}" var="record">
                <tr>
                    <td>{!record.Name}</td>
                    <td>{!record.Amount}</td>
                </tr>
            </aura:iteration>
        </tbody>
    </table>
</aura:application>

opportunityController.js

({
    doInit: function(component, event, helper) {
        var load = component.get("c.loadOpportunityRecords");
        load.setCallback(this, function(result) {
            component.set("v.records", result.getReturnValue());
            component.set("v.sortAsc", true);
            helper.sortBy(component, "Name");
        });
        $A.enqueueAction(load);
    },
    sortByName: function(component, event, helper) {
        helper.sortBy(component, "Name");
    },
    sortByAmount: function(component, event, helper) {
        helper.sortBy(component, "Amount");
    }
})

opportunityHelper.js

({
    sortBy: function(component, field) {
        var sortAsc = component.get("v.sortAsc"),
            sortField = component.get("v.sortField"),
            records = component.get("v.records");
        sortAsc = field == sortField? !sortAsc: true;
        records.sort(function(a,b){
            var t1 = a[field] == b[field],
                t2 = a[field] > b[field];
            return t1? 0: (sortAsc?-1:1)*(t2?-1:1);
        });
        component.set("v.sortAsc", sortAsc);
        component.set("v.sortField", field);
        component.set("v.records", records);
    }
})